GEOMETRY  OF  GREEK  VASES 


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MUSEUM  OF  FINE  ARTS,  BOSTON 

COMMUNICATIONS  TO  THE  TRUSTEES,  V 


GEOMETRY 

OF 

GREEK  VASES 


ATTIC  VASES  IN  THE  MUSEUM  OF  FINE  ARTS 
ANALYSED  ACCORDING  TO  THE  PRINCIPLES  OF 
PROPORTION  DISCOVERED  BY  JAY  HAMBIDGE 


BY 

L.  D.  CASKEY 

CURATOR  OF  CLASSICAL  ANTIQUITIES 


BOSTON 

MCMXXII 


COPYRIGHT,  1922,  BY  MUSEUM  OF  FINE  ARTS,  BOSTON 
ALL  RIGHTS  RESERVED 


,tffY  CENTER 
LIBRARY 


NOTE 


Beginning  in  1904  the  Museum  printed  four  volumes  of  Communica- 
tions, consisting  of  studies  on  problems  which  confronted  the  Trustees 
on  the  occasion  of  the  erection  of  the  present  building,  viz. : 

I.  Papers  on  the  building  and  installation  of  a Museum  of 
Art,  1904. 

II.  The  Exhibition  of  Casts;  Museum  Methods,  1904. 

III.  The  Museum  Commission  in  Europe,  1905. 

IV.  The  Experimental  Gallery,  1906. 

These  volumes  have  long  been  out  of  print.  The  Museum  has  met  fre- 
quent demands  for  them  by  the  use  of  lending  copies,  which  are  available 
for  periods  of  three  months  or  longer  if  not  otherwise  in  demand. 

In  resuming  the  series  of  Communications  the  Museum  undertakes 
again  the  publication  of  studies  by  officers  of  the  Museum  on  subjects 
connected  with  the  branches  of  art  or  of  Museum  activity  which  the 
writers  represent.  A quarto  form  has  been  adopted  as  most  convenient 
and  most  suitable  for  illustration.  The  series  will  not  be  a periodical 
publication,  but  studies  will  be  printed  occasionally  and  without  regard 
to  their  compass. 


PREFACE 


In  Mr.  Hambidge’s  book,  Dynamic  Symmetry:  The  Greek  Vase,  a remarkable 
connection  is  traced  between  the  proportions  of  Attic  vases  and  those  of  certain 
rectangles  derived  from  the  square  by  a simple  process  which  was  familiar  to 
Greek  geometers.  Mr.  Hambidge  finds  that  if  the  over-all  proportion  of  a vase, 
that  is,  the  ratio  existing  between  its  height  and  greatest  width,  is  expressible  in 
terms  of  one  of  these  rectangles,  then  the  heights  and  widths  of  all  its  parts  can  be 
expressed  in  terms  of  that  rectangle,  and  of  no  other.  The  vase  possesses  sym- 
metry in  the  sense  that  all  its  elements  are  commensurable,  the  common  factor  or 
coordinating  principle  being  a rectangle  whose  sides  illustrate  one  of  the  simple 
ratios  1 :\/l,  1 :y/2,  1 -\/S,  l:\/5.  Except  for  the  first,  these  proportions  are  com- 
mensurable in  square,  but  not  in  line;  they  can  be  studied  in  terms  of  geometry, 
but  not  in  terms  of  arithmetic. 

Having  observed  this  phenomenon  not  only  in  Attic  pottery  and  other  prod- 
ucts of  the  minor  arts  having  an  architectonic  character,  but  also  in  Greek 
tempos,  notably  the  Parthenon,  Mr.  Hambidge  has  advanced  the  theory  that 
Greek  artistic  design  was  based  on  geometrical  principles.  In  the  absence  of  clear 
and  reliable  literary  evidence  this  conclusion  cannot  be  rigorously  proved,  or  dis- 
proved. But  the  coincidences  which  Mr.  Hambidge  has  brought  to  light  are  too 
striking  to  be  ignored;  and  progress  towards  a solution  of  the  problem  they  pre- 
sent can  be  made  only  by  a thorough  study  of  as  many  extant  monuments  as 
possible. 

In  this  book  I have  tried  to  present  in  an  intelligible  form  all  the  evidence  af- 
forded by  the  collection  of  Attic  black-figured  and  red-figured  pottery  in  the 
Museum  of  Fine  Arts.  With  some  exceptions,  which  are  noted  in  the  introduc- 
tion, I have  investigated  every  example  which  lent  itself  to  this  sort  of  analysis, 
and  have  arranged  the  material  in  the  form  of  a catalogue,  in  which  every  type  of 
vase  in  the  collection,  except  the  rhyton,  the  phiale  and  the  plate,  is  represented. 
The  proportions  are  expressed  in  two  ways  — by  drawings  and  by  tables  of 
ratios.  The  former  are  reproductions,  usually  much  reduced,  of  full-size  eleva- 
tions. More  than  three-fifths  of  the  vases  have  been  measured  and  drawn  by  my- 
self. The  rest  of  the  drawings  have  been  executed  by  Mr.  E.  G.  Go  wen  under  my 
supervision;  I have  checked  his  measurements  in  every  case,  and  am  responsible 
for  their  accuracy.  The  outline  of  each  vase  is  enclosed  in  a rectangle  belonging  to 
one  of  the  systems  of  dynamic  symmetry,  the  margin  of  error  allowed  averaging 
less  than  one  millimetre.  The  interrelation  of  details  is  shown  by  subdivisions  of 
the  containing  rectangles  and  by  intersections  of  diagonals.  It  has  often  proved 
impossible  to  analyse  vases  of  complicated  structure  without  using  a confusingly 


[ vii] 


PREFACE 


great  number  of  lines.  This  might  have  been  avoided  by  publishing  several  draw- 
ings of  each  vase,  or  by  including  only  those  proportions  which  could  be  most 
simply  expressed.  The  former  alternative  has  been  rejected  from  motives  of 
economy;  the  latter  from  a desire  to  achieve  thoroughness,  even  at  the  expense  of 
clearness.  In  a few  cases  I have  shown  some  of  the  more  striking  coincidences  by 
means  of  subsidiary  diagrams.  The  text  accompanying  each  drawing  gives  the 
chief  dimensions  of  the  vase,  with  a description  and  bibliography  sufficient  to 
identify  it,  a brief  explanation  of  the  geometrical  analysis,  and  a table  of  ratios  in 
which  the  proportions  are  expressed  arithmetically — usually  in  the  form  of  inde- 
terminate fractions  carried  to  three  or  four  decimal  places.  These  ratios  are  given 
chiefly  because  of  their  practical  usefulness  as  labels.  In  the  introduction  I have 
explained  the  rectangles  used  in  the  analyses,  the  methods  of  subdivision,  and  the 
resulting  ratios.  I have  also  submitted  some  statistics  regarding  the  margin  of 
error  allowed,  and  the  number  of  occurrences  of  the  more  important  proportions. 

The  title  of  the  book  was  suggested  by  a remark  of  M.  Edmond  Pottier,  which 
is  quoted  on  page  32  of  the  introduction.  Whether  arithmetic  or  geometry  can  be 
of  aid  to  the  artist  in  achieving  symmetry,  i.  e.,  harmony  of  proportions,  is  open  to 
question.  But,  if  either  is  to  be  used,  there  can  be  no  doubt  that  geometry  is  the 
simpler  instrument.  Artists  cannot  dispense  with  the  use  of  linear  units  for  certain 
practical  purposes.  A few  proportions,  such  as  1:1,  1:2,  can  be  determined  as 
easily  by  arithmetic  as  by  geometry.  But  the  use  of  numbers  if  carried  farther, 
leads  either  to  monotonous,  stereotyped  designs,  or  to  hopeless  confusion.  For 
example,  the  skyphos,  no.  104,  has  dimensions  which  can  be  expressed  by  the 
numbers  8,  15,  11,  5,  7.  It  would  take  a Pythagoras  to  explain  the  proportional 
value  of  these  numbers,  whereas  the  geometrical  analysis  of  the  vase  is  readily 
understood  and  appreciated  by  a child.  Many  of  the  analyses  in  this  book  are 
equally  simple;  many  others  are  much  more  complicated.  The  cups,  nos.  132  and 
136,  may  be  cited  as  extreme  cases,  the  former  telling  in  favor  of  the  theory,  the 
latter  against  it.  A judgment  as  to  its  probability  or  improbability  must  take 
account  of  all  the  examples.  My  own  position  may  be  stated  baldly  as  follows: 

(1)  The  coincidences  are  in  so  many  cases  so  accurate,  simple  and  logical  that  I 
find  it  less  difficult  to  believe  them  due,  in  part  at  least,  to  conscious  design,  than' 
to  instinctive  obedience  to  a mysterious  aesthetic  law,  or  to  mere  accident. 

(2)  The  proportion  obtained  by  dividing  a line  in  extreme  and  mean  ratio,  which 
plays  an  important  part  in  Euclidean  geometry,  has  for  ages  been  recognised  as  an 
ever-recurring  phenomenon  in  artistic  design.  It  has  been  called  by  various  names 
— divine  proportion,  golden  section,  ratio  of  Phidias,  and  the  like;  and  it  has 
been  studied  in  many  ways.  By  considering  it  as  an  area,  rather  than  as  a divi- 
sion of  a line,  and  by  emphasising  its  relation  to  the  \/5  rectangle,  Mr.  Ham- 
bidge  has  immensely  simplified  the  problem  of  investigating  its  significance.  He 
has  not  only  made  it  easier  to  detect  the  occurrences  of  this  proportion;  he  has 
revealed  its  more  important  function  as  a coordinating  principle  in  designs  in 
which  it  does  not  itself  necessarily  occur.  For  this  reason  alone  Mr.  Hambidge’s 
work  is  of  permanent  value. 


[ viii  ] 


PREFACE 


A criticism  of  Dynamic  Symmetry,  published  by  Professor  Rhys  Carpenter  in 
the  American  Journal  of  Archaeology,  'XXV,  1921,  pp.  18-36,  concludes  that 
“a  priori,  the  probabilities  are  all  against  its  being  true.”  A different  explanation 
of  the  coincidences  must  therefore  be  sought.  Mr.  Carpenter  falls  back  on  the 
theory  that  Athenian  potters  made  the  dimensions  of  their  vases  conform  to 
units  of  a foot-rule.  This  would  be  a simple  and  legitimate  solution,  if  it  could  be 
shown  to  fit  the  evidence  satisfactorily.  It  was  rejected  by  me  (as  it  had  been  by 
Mr.  Hambidge)  after  a more  thorough  study  than  appears  to  have  been  devoted 
to  it  by  Mr.  Carpenter.  A reply  to  his  article,  which  I prepared  at  Mr.  Ham- 
bidge’s  suggestion,  has  been  declined,  unread,  by  the  editor  of  the  Journal.  Since 
access  to  this  periodical  has  been  denied  me,  I have  revised  the  concluding  para- 
graphs of  the  following  introduction  in  the  hope  of  throwing  some  light  on  the 
question  at  issue. 

I desire  to  express  my  thanks  to  the  Trustees  of  the  Museum  for  publishing  this 
book  in  the  series  of  Communications,  and  among  them  especially  to  Dr.  Denman 
W.  Ross  for  his  interest  in  the  work,  and  for  furnishing  the  services  of  a 
draughtsman. 

While  the  investigation  was  in  progress  I had  the  privilege  of  consulting  Mr. 
Hambidge  almost  daily.  The  determination  of  the  over-all  proportions  of  the 
vases  is  due  to  him  in  most  cases,  the  geometrical  analysis  of  details  in  many 
cases.  I am  indebted  to  him  above  all  for  explaining  his  theory  to  me  in  all  its 
phases,  and  for  giving  me  access  to  his  vast  collection  of  unpublished  material  — 
the  result  of  minute  and  laborious  researches  carried  on  for  many  years.  That 
Mr.  Hambidge  possesses  the  imaginative  insight  to  construct  daring  and  sugges- 
tive hypotheses  is  apparent  to  all  who  read  his  book;  only  those  who  have  worked 
with  him  can  fully  appreciate  his  industry  in  collecting  facts,  his  accuracy  in 
recording  them,  his  cautious  and  critical  attitude  towards  them  as  evidence. 


C ix  ] 


CONTENTS 


Preface vii 

Introduction 1 

Analyses  of  Vases: 

Amphora  (1-44)  35 

Pelike  (45-50) 83 

Stamnos  (51-56) 91 

Hydria-Kalpis  (57-69) 101 

Deinos-Krater  (70-83) 116 

PsYKTER  (84) 131 

Oinochoe-Olpe  (85-100) 132 

Skyphos  (101-117) 148 

Cup  with  Impressed  Decoration  (118) 159 

Kantharos  (119-123) 160 

Kylix  (124-162) 167 

Lekythos  (163-180) 209 

Pyxis  (181-182) • • • • 226 

Perfume  Vase  (183-185) 230 


[xi] 


GEOMETRY  OF  GREEK  VASES 


INTRODUCTION 

No  attempt  is  made  in  this  book  to  give  a new  statement,  a resume,  or  a criti- 
cism of  Mr.  Hambidge’s  revolutionising  theory  in  regard  to  Greek  design.  Study 
of  his  book,  Dynamic  Symmetry:  The  Greek  Vase,  and  of  his  periodical,  The 
Diagonal,  is  indispensable  to  the  understanding  of  the  principles  of  proportion 
which  he  has  discovered.  In  the  present  work  these  principles  are  accepted  as  a 
starting  point  and  applied  to  the  examples  of  Attic  pottery  in  the  Museum.  All 
that  is  required  by  way  of  introduction  is  an  explanation,  in  as  simple  language  as 
possible,  of  the  rectangles  which  are  used  in  analysing  the  vases. 

The  first  point  to  be  grasped  — and  its  importance  cannot  be  overemphasised 
— is  that  Mr.  Hambidge  has  opened  up  an  entirely  new  method  of  approach  to 
the  problem  of  determining  the  proportions  of  Greek  works  of  art.  And  this 
method,  though  at  first  sight  strange  to  us  of  the  modern  world,  is  just  the  one 
which  the  Greeks  might  naturally  be  expected  to  have  employed.  Since  the  time 
of  Vitruvius  attempts  to  analyse  Greek  proportions  have  generally  been  based 
on  units  of  linear  measurement.  And  every  one  will  admit  that  the  results  ob- 
tained— -whether  the  unit  experimented  with  was  a Greek  foot,  or  a modulus 
taken  from  the  object  analysed  — have  been  meagre,  and  for  the  most  part  un- 
convincing. What  the  investigators  have  failed  to  take  into  account  is  that  the 
science  of  numbers  was  still  in  its  infancy  during  the  culminating  period  of  Greek 
art,  whereas  geometry  was  a highly  developed  science  long  before  the  days  of 
Euclid.  If  the  Greeks  consciously  employed  any  system  of  proportions,  it  is 
a priori  more  probable  that  they  based  it  on  relations  of  areas  rather  than  on 
relations  of  lines.  In  other  words  they  would  have  used  geometry  rather 
than  arithmetic. 

Having  realised  this  Mr.  Hambidge  made  the  startling  discovery  that  the 
proportions  found  in  Greek  works  of  art,  which  are  in  perhaps  nine  cases  out  of 
ten  incommensurable  in  terms  of  a linear  unit,  or  modulus,  can  in  the  large 
majority  of  cases  be  accurately  and  intelligibly  expressed  in  terms  of  areas  possess- 
ing certain  clearly  definable  properties. 

Nine  (and  possibly  a few  more)  vases  studied  in  this  book  have  proportions 
which  can  be  expressed  exactly  in  whole  numbers.  The  two  deinoi  (nos.  70,  71), 
for  example,  have  a relation  of  height  to  diameter  of  4 to  5.  It  is  obvious  that  this 
proportion  can  also  be  expressed  geometrically  by  enclosing  the  vases  in  a rec- 
tangle with  sides  in  the  ratio  4 : 5.  Such  a rectangle  is  made  up  of  twenty  squares. 
And  the  proportions  of  details  can  be  expressed  in  simple  subdivisions  of  these 
squares.  The  two  vases  illustrate  what  Mr.  Hambidge  has  called  static  symmetry. 


GEOMETRY  OF  GREEK  VASES 


A more  complicated  example  of  static  symmetry  is  furnished  by  the  lekythos,  no. 
167.  Its  height  equals  2§  times  its  diameter.  If  the  diameter  is  regarded  as  unity 
the  proportions  are  as  follows  (column  A) : 


Height 

Height  to  shoulder 

Height  of  lip,  neck  and  shoulder 

Height  of  lip 

Height  of  neck 

Height  of  shoulder 

Diameter 

Diameter  of  lip 

Diameter  of  bottom  of  body 
Diameter  of  foot 


A 

B 

2'i 

150  units 

If 

105 

U 

3 

4 

45 

u 

10 

18 

(6 

3_ 

10 

18 

<( 

3_ 

20 

9 

a 

1 

60 

u 

3 

36 

a 

1 

3 

20 

u 

3 

4 

45 

u 

In  order  to  express  all  these  proportions  in  whole  numbers  the  diameter  of  the 
vase  must  be  divided  into  no  less  than  60  units.  The  proportions  in  terms  of  these 
units  are  given  in  column  B in  the  table.  It  is  of  course  inconceivable  that  a 
potter  should  have  used  a unit  2 mm.  long  in  working  out  the  proportions  of  a 
vase,  or  that  he  should  have  used  the  diameter  as  a unit,  and  have  divided  it  into 
halves,  thirds,  quarters,  fifths,  tenths,  and  twentieths.  The  alternatives  are  to 
suppose  that  these  relations  of  part  to  whole  and  of  part  to  part  were  arrived  at 
unconsciously,  or  that  they  were  worked  out  by  a geometrical  construction  similar 
to  that  shown  on  page  213.  Whichever  hypothesis  is  preferred,  the  fact  remains 
that  the  proportions  of  the  lekythos  can  be  more  clearly  and  simply  expressed  in 
terms  of  squares  than  in  terms  of  units  of  length. 

The  vast  majority  of  vases  investigated  have  proportions  which  cannot  be 
accurately  expressed  either  in  linear  units  or  in  squares,  but  which  can  be  clearly 
analysed  in  terms  of  rectangles  derived  geometrically  from  the  square.  These 


V3 


<  V4 

<  V5 

Diagram  I.  The  root  rectangles 
[2] 


INTRODUCTION 


vases  illustrate  what  Mr.  Hambidge  has  called  dynamic  symmetry.  The  rec- 
tangles of  dynamic  symmetry  fall  into  two  classes:  (1)  Those  derived  from  the 
diagonal  of  a square;  (2)  those  derived  from  the  diagonal  of  two  squares.  The 
second  class  is  of  course  related  to  the  first,  but  possesses  some  remarkable  quali- 
ties which  justify  its  being  treated  separately.  The  first  class  is  composed  of  the 
“root-rectangles”  and  their  derivatives.  The  root-rectangles  are  those  whose 
short  side  is  unity  and  whose  long  side  is  equal  to  \/2,  \/3,  \/4,  \/5,  etc.,  respec- 
tively. A simple  method  of  constructing  these  rectangles  is  shown  in  diagram  I. 
The  diagonal  of  a square  whose  side  is  unity  equals  \/2,  which  can  be  expressed 
approximately  by  the  irrational  fraction  1.4142 A \/2  rectangle  is  con- 

structed by  using  the  diagonal  of  the  generating  square  as  the  radius  of  a circle. 
This  circle  cuts  the  base  of  the  square  produced  so  as  to  fix  the  length  y/2.  A per- 
pendicular from  this  point  to  the  top  of  the  square  produced  completes  a \/2  rec- 
tangle. The  diagonal  of  a \/2  rectangle  equals  \/3  (1 .732 . . . . ) . The  diagonal  of  a 
\/3  rectangle  equals  v/4  (2.000).  The  diagonal  of  a -\/4  rectangle  equals  \/5 
(2.236.  . . .),  and  so  on.  Each  root-rectangle  is  constructed  in  the  same  way  from 
the  previous  one.  For  the  purposes  of  this  investigation  the  \/ 4 rectangle  may  be 
disregarded  since  it  is  composed  of  two  squares.  The  root-rectangles  beyond  \/ 5 
are  also  omitted  from  consideration.  Of  the  vases  here  published  nine  or  more  are 
based  on  the  square,  eighteen  on  the  \/2  rectangle,  six  on  the  \/3  rectangle.  The 
remainder,  some  136  in  all,  have  proportions  based  on  the  \/5  rectangle,  or  on  a 
rectangle  intimately  related  to  it,  which  remains  to  be  described. 

It  will  readily  be  seen  that  the  \/5  rectangle  is  derived  from  the  diagonal  of 
two  squares,  since  a/4  = 2.  Diagram  II  shows  a second  method  of  describing  the 
v/5  rectangle.  If  a square  with  sides  equal  to  unity  is  bisected  vertically,  each 
half  is  composed  of  two  squares  with  sides  equal  to  | or  .500.  The  diagonal  of 
half  the  larger  square  may  thus  be  regarded  as  the  diagonal  of  two  squares,  and 


its  length  is  or  1.118 A semicircle  with  the  centre  of  the  base  of  the 

" . -\/5 

large  square  as  centre  and  the  diameter  of  half  that  square,  — , as  radius  will 

A 


Diagram  II.  The  V 5 rectangle 


GEOMETRY  OF  GREEK  VASES 


have  a diameter  equal  to  V 5.  And  a rectangle  with  that  diameter  as  base  and 
unity  as  height  will  be  a a/5  rectangle.  This  rectangle  is  composed  of  a square 


flanked  by  rectangles  with  unity  as  height  and 

K 2.236  - 1.000  R1Q 

be  expressed  as , or  .618 


V5  “ 1 

— - — as  width, 


This  may  also 


At  this  point  it  is  convenient  to  consider  the  methods  of  obtaining  the  recip- 
rocal of  a given  rectangle.  The  reciprocal  is  a rectangle  of  the  same  shape  as  the 
original  rectangle,  and  with  its  long  side  equal  to  the  short  side  of  the  original 
rectangle.  A reciprocal  may  be  obtained  by  drawing  a diagonal  of  the  rectangle 
and  erecting  a perpendicular  to  the  diagonal  at  one  of  its  ends.  The  point  of  inter- 
section of  this  perpendicular  with  the  opposite  side  of  the  rectangle  produced 
determines  the  width  of  the  reciprocal.  In  diagram  III  the  rectangle  B C is  the 


Diagrams  III  and  IV.  Rectangles  and  their  reciprocals 


reciprocal  of  the  rectangle  A B.  In  this  case  the  reciprocal  is  added  to  the  end  of 
the  rectangle.  A reciprocal  may  be  cut  off  from,  or,  as  the  Greeks  expressed  it, 
applied  to  a rectangle,  by  drawing  the  diagonal  of  the  rectangle,  and  dropping  a 
perpendicular  upon  it  from  one  of  the  opposite  angles.  This  perpendicular  pro- 
duced intersects  the  opposite  side  so  as  to  determine  the  width  of  the  reciprocal. 
In  diagram  IV  the  rectangle  C D is  the  reciprocal  of  the  rectangle  A B. 

The  root-rectangles  and  their  reciprocals  have  a relation  to  one  another  which 
is  of  importance  in  the  geometric  analysis  of  proportions  based  on  these  rec- 
tangles. If  a reciprocal  is  applied  to  a y/2  rectangle,  the  remaining  or  excess 


\ 

V3 

\ 

Diagram  V.  The  root  rectangles  and  their  reciprocals 


area  is  also  a reciprocal  of  the  rectangle.  In  other  words  the  \/2  rectangle  is 
composed  of  two  \/2  rectangles.  Similarly  the  -y/3  rectangle  is  composed  of  three 
v/3  rectangles,  the  \/4  rectangle  of  four  \/4  rectangles,  the  y/b  rectangle  of  five 
y/5  rectangles,  and  so  on.  Cf.  diagram  V. 

Returning  to  the  y/5  rectangle  subdivided,  as  in  diagram  II,  into  a square 


flanked  by  two  rectangles  of  the  proportions 


V5-1 


or  1 : .618,  it  is  obvious 


[4  ] 


2 


INTRODUCTION 


that  in  diagram  VI  the  diagonal  of  the  area  A+B  meets  the  diagonal  of  the  area  C 
at  a right  angle,  since  all  angles  in  a semicircle  are  right.  The  rectangle  C is  thus 
seen  to  be  a reciprocal  of  the  rectangle  A-\-B.  The  rectangle  A,  being  equal  to  C, 
is  also  a reciprocal  of  the  rectangle  A+B.  The  rectangle  A+B  is  thus  seen  to 
have  the  interesting  property  that,  if  a reciprocal  (A)  be  applied  to  it,  the  excess 
area  (B)  is  a square.  A reciprocal  applied  to  the  reciprocal  again  leaves  a square, 
and  so  on.  If  this  process  is  continued  indefinitely  a series  of  squares  is  produced, 


Diagram  VI.  The  V 5 rectangle  and  the 
rectangle  of  the  whirling  squares 


which  continually  decrease  in  size  and  revolve  about  the  point  of  intersection  of 
the  diagonal  with  its  perpendicular  in  a logarithmic  spiral  to  infinity.  Cf.  dia- 
gram VII.  Because  of  this  property  Mr.  Hambidge  has  named  this  rectangle  the 
rectangle  of  the  whirling  squares. 

The  application  of  the  reciprocal  to  a whirling  square  rectangle  divides  the 
base  of  the  rectangle  in  extreme  and  mean  ratio.  That  is,  the  smaller  part  is  to 
the  larger  part  as  the  larger  part  is  to  the  whole  line.  .618:1.000  = 1.000  A. 618,  or 

expressing  it  in  terms  of  \/5 : — ~ 1 = 1 : - • This  division  of  a line  is  of 

2 2 

course  the  well-known  divine  section  or  divine  proportion  of  Paccioli,  Kepler,  and 
Leonardo  da  Vinci,  more  generally  called  in  recent  years  the  golden  section. 

Eleven  vases  published  in  this  book  are  contained  in  the  whirling  square  rec- 
tangle, i.  e.,  their  diameter  is  to  their  height  in  the  ratio  of  1:1.618.  Cf.  diagram 


[5  ] 


GEOMETRY  OF  GREEK  VASES 


VIII  in  which  ten  of  these  vases  are  shown.  In  eight  cases  the  rectangle  encloses 
the  complete  vase ; in  three  it  contains  the  vase  without  its  handles,  The  eleventh 
example,  no.  93,  is  omitted  because  it  is  practically  a duplicate  of  the  oinochoe 
no.  92. 

The  whirling  square  rectangle  is  one  of  the  most  common  but  by  no  means  a 
predominating  shape.  Its  aesthetic  significance  as  Mr.  Hambidge  has  observed, 
lies  rather  in  its  value  as  a coordinating  factor.  A very  large  proportion  of  the 
vases  studied  are  contained  in  rectangles  derived  in  simple  ways  from  the  whirl- 
ing square  rectangle.  Some  of  these  will  be  described  below.  In  many  examples 


w 

w 

86 


87  92  97 

Diagram  VIII.  Examples  of  the  rectangle  1.618  = .618 


185 


the  ratio  .618  occurs  as  the  proportion  of  some  detail.  About  thirty  instances  of 
this  have  been  noted,  and  it  is  probable  that  a more  exhaustive  study  would 
reveal  still  others.  But  the  occurrences  of  this  rectangle  as  the  containing  area  of 
the  whole  or  a portion  of  a vase,  or  as  the  proportion  of  a detail  by  no  means 
exhaust  its  possibilities.  It  can  itself  be  subdivided  in  various  ways,  producing 
new  proportions  related  to  the  generating  form,  which  appear  repeatedly  in 
the  analyses  of  the  vases  based  on  this  system.  Some  of  the  more  common 
subdivisions  of  the  whirling  square  rectangle  are  illustrated  in  the  following  dia- 
grams by  simple  geometrical  constructions.  They  should  be  studied  in  connec- 
tion with  the  numerical  ratios,  which  will  be  found  extremely  useful,  if  not 
indispensable,  in  memorising  them. 

Diagram  IX  represents  a whirling  square  rectangle  placed  horizontally,  the 
width  being  regarded  as  unity,  the  height  as  .618.  By  applying  squares  at  either 
end,  and  by  applying  squares  at  the  bottom  of  each  reciprocal  the  area  may  be 
divided  horizontally  into  two  rectangles,  the  upper  one  of  which  is  composed  of  a 
square  flanked  by  horizontal  whirling  square  rectangles,  while  the  lower  one  is 
composed  of  a vertical  whirling  square  rectangle  flanked  by  squares.  The  height 
of  the  upper  rectangle  is  .236,  that  of  the  lower  .382.  The  former  is  the  reciprocal 


INTRODUCTION 


of  4.236,  the  latter  of  2.618.  All  these  ratios — .236,  .382,  .618,  1.000,  1.618, 
2.618,  4.236  — are  terms  in  a continuous  proportion. 

As  many  as  forty-five  occurrences  of  the  proportion  .382  have  been  noted  in 
this  book.  The  rectangle  .382  = 2.618  will  be  further  considered  below.  The  ratio 
.236  occurs  at  least  thirty-one  times. 

In  diagram  X a,  perpendiculars  dropped  from  the  centres  of  the  applied  squares 
divide  the  base  into  the  ratios  .309 +.382 +.309.  Perpendiculars  from  the  centres 

tf~ [\  71 

CO 

cm  y \ 

> + ‘r-r 


ca 

60 


< .382 A— .236 — * .382 * 

Diagram  IX.  Subdivisions  of  the  whirling 
square  rectangle 

of  the  reciprocals  divide  the  base  into  .191  + .618  + .191.  Simpler  geometrical 
expressions  of  these  ratios  are  given  in  diagram  X,  b and  c.  It  is  worth  noting 
that  .309  is  half  of  .618  and  that  .191  is  half  of  .382. 

In  diagram  XI  the  process  of  applying  squares  is  carried  a step  farther,  and 
produces  the  ratios  .236+. 528+. 236,  and  .146+.708+.146.  Many  of  the  vases 
analysed  show  the  proportions  .528  and  .708.  The  latter  equals  .236X3. 

The  ratio  .5669  which  occurs  several  times  is  obtained  from  the  intersection  of 
the  diagonal  of  half  a whirling  square  rectangle  with  the  diagonal  of  the  reciprocal, 
as  shown  in  diagram  XII. 

Diagram  XIII  shows  the  relation  of  the  important  ratio  .4472  (the  reciprocal  of 
\/5,  or  2.236)  to  the  whirling  square  rectangle.  If  a horizontal  line  is  drawn 


< .309 * .382 * 309 » < .309 *• .382  — * .309 * <-.191  — * .618 x— .199— * 

<-.191— « .618 x-,191— » 

a be 

Diagram  X.  Subdivisions  of  the  whirling  square  rectangle 


through  the  points  of  intersection  of  the  diagonals  of  a whirling  square  rectangle 
with  the  diagonals  of  its  reciprocals,  the  area  below  this  line  is  a \/5  rectangle. 
Perpendiculars  dropped  from  these  points  of  intersection  upon  the  base  divide 
the  v/5  rectangle  into  a square  and  two  whirling  square  rectangles.  Since  the 


GEOMETRY  OF  GREEK  VASES 


base  is  unity,  the  height  of  the  y/5  rectangle,  and  the  side  of  its  central  square 
have  the  ratio  .4472,  the  reciprocal  of  2.236,  or\^5.  The  base  is  divided  into  the 


F""~  1 T“ 

' / 

• . ' / 

/ 

» 1 \ / 

/ 

/ 

1 i //  1 

V / 

• \ ' 

/ I •' 

• ! / v 

' 1 \ 

/ 1 X 

, 1 ' 

| :/  ; \ 

/ 1 

X < , N 

/ 1 

\ 1 /'I  ■ \ 

lil i 1 1 

146 -* .708 *-.l46-f  * — 5 .5669 

Diagrams  XI  and  XII.  Subdivisions  of  the  whirling  square  rectangle 


ratios  .2764+. 4472+. 2764.  The  diagram  also  accounts  for  the  ratios  .1382,  .7236, 
and  .1708.  The  ratio  .4472  occurs  at  least  thirty  times  in  the  vases  analysed. 

In  diagram  XIV  the  y/ 5 rectangle  is  shown  with  two  whirling  square  rectangles 
in  the  centre  and  half  a square  on  each  side.  This  supplies  the  ratios  .2236  and 
.5528. 

These  examples  will  suffice  to  show  the  importance  of  the  whirling  square 
rectangle  as  a containing  area,  and  as  a coordinating  factor  for  determining  other 


t <•: 7. v v 

0O  I ' / ; ; x ' 1 

Px  . / . -V  i 


v x 

7 

\ 

- ' 

\ / 

\ f 

y. 

' 

f.  i 

, i \ 

' 1 V' 
/ t \ 

L 

• ! s x %\ 

«— .2764 — * .4472 *— .2764  — » 

*1382-* .7236 *-.l382-»- 


Diagram  XIII 


Diagram  XV.  The  rectangle  1.236  = .809 


rectangles  and  proportions  occurring  repeatedly  in  Attic  vases.  It  remains  to 
consider  some  of  the  more  common  areas  derived  from  the  whirling  square 
rectangle  and  the  y/5  rectangle. 


INTRODUCTION 


Perhaps  the  commonest  shape  of  all  is  that  composed  of  two  whirling  square 
rectangles  placed  side  by  side,  which  can  also  be  regarded  as  half  a whirling 


55  72  91  95  107 


109  110  111,  112,  113,  115  120  185 

Diagram  XVI.  Examples  of  the  rectangle  1.236  = .809 


square  rectangle.  Cf.  diagram  XV.  The  ratio  of  this  rectangle  is  1.236,  or  .618  X 2. 
Its  reciprocal  is  .809,  or  1. 618-1- 2.  Diagram  XVI  shows  some  of  the  vases  which 
are  contained  in  this  area.  Cf.  also  the  pelike  no.  46,  the  red-figured  skyphoi  nos. 
Ill,  112,  115,  the  black-figured  amphorae  nos.  18,  19  (up  to  the  shoulder),  the 
stamnos  no.  50  (up  to  the  shoulder),  the  stamnos  no.  56  (omitting  the  handles), 
the  black-figured  krater  no.  73  (omitting  the  handles),  the  bowl  of  the  same 
krater,  the  red-figured  krater  no.  76  (up  to  the  lip),  twenty-one  examples  in  all. 
The  ratio  .809  also  occurs  a number  of  times. 


The  rectangle  composed  of  three  whirling  square  rectangles  placed  side  by 
side  (ratio:  1.854;  reciprocal:  .5394)  occurs  seven  times.  Cf.  diagram  XVII,  and 
nos.  37  and  88. 


GEOMETRY  OF  GREEK  VASES 


The  rectangle  2.854  contains  the  black-figured  kylix,  no.  132,  with  its  handles, 
and  the  red-figured  kylix,  no.  161,  without  its  handles.  The  latter,  with  its 
handles  included,  fits  the  rectangle  3.854. 

Four  whirling  square  rectangles  placed  side  by  side  make  an  area  with  the 
ratio:  2.472  (reciprocal:  .4045).  The  two  red-figured  kylikes,  nos.  140,  141,  are 
contained  in  this  shape,  if  the  handles  are  omitted.  Cf.  diagram  XVIII. 


140,  141 

Diagram  XVIII 
The  rectangle  2.472  = .4045 


138,  139,  140 

Diagram  XIX 
The  rectangle  3.090  = .3236 


Diagram  XX.  The  rectangle  3.236  = .309 

The  area  composed  of  five  whirling  square  rectangles  placed  side  by  side 
(ratio:  3.090;  reciprocal:  .3236)  occurs  as  the  over-all  shape  of  the  three  red- 
figured  kylikes,  nos.  138-140.  Cf.  diagram  XIX. 

The  areas  just  described  are  obtained  by  placing  vertical  whirling  square  rec- 
tangles side  by  side.  If  two  horizontal  whirling  square  rectangles  are  placed  side 


by  side  the  resulting  area  has  the  ratio  3.236  (reciprocal:  .309).  This  is  one  of  the 
two  most  common  containing  areas  of  red-figured  kylikes.  Cf.  diagram  XX,  nos. 
141-147. 

This  rectangle  added  to  a square  produces  an  area  with  the  ratio  1.309  (recip- 
rocal: .764  = .382X2),  which  is  found  several  times.  Cf.  diagram  XXI. 

The  same  rectangle  added  to  three  squares  (3.309)  produces  the  containing 
area  of  the  lekythos,  no.  175. 


[ 10  ] 


INTRODUCTION 


Three  pairs  of  whirling  square  rectangles  placed  one  above  the  other  produce 
an  area  with  the  ratio  1.0787,  more  simply  expressed  in  terms  of  its  reciprocal, 
.927  = .309X3.  Three  examples  are  shown  in  diagram  XXII.  The  hydria  no.  55 
and  the  kalpis  no.  65  are  also  perhaps  contained  in  this  shape. 

The  ratio  .382,  which  is  the  reciprocal  of  2.618,  was  noted  above  as  being  one 
of  the  most  common.  Its  use  as  a containing  rectangle  seems  to  have  been  com- 


paratively rare.  Four  red-figured  kylikes  without  their  handles  (nos.  145-147, 
155,  and  the  lid  of  the  pyxis,  no.  181)  illustrate  this  shape.  Cf.  diagram  XXIII. 

This  ratio  (.382)  combined  with  .809  produces  the  rectangle  1.191  which  occurs 
at  least  three  times.  Cf.  diagram  XXIV,  and  no.  51. 

The  ratio  .382  added  to  a square  produces  one  of  the  most  important  areas  of 
dynamic  symmetry  — the  rectangle  1.382  (reciprocal : .7236).  It  has  an  intimate 


Diagram  XXIII.  Examples  of  the  rectangle  2.618  = .382 


relation  to  the  equally  important  area  1.4472,  composed  of  a square  and  a \/5 
rectangle,  which  will  be  discussed  below.  Two  black-figured  amphorae,  nos.  4,  5 ; 
a pelike,  no.  49,  a kalpis,  no.  69,  a black-figured  skyphos,  no.  104,  a kantharos, 
no.  123,  two  black-figured  kylikes,  nos.  130,  131,  and  a black-figured  perfume 
vase,  no.  183,  illustrate  this  shape.  Cf.  diagram  XXV. 


Diagram  XXIV.  Examples  of  the  rectangle  1.191  = .8396 


[ 11  ] 


GEOMETRY  OF  GREEK  VASES 


Diagram  XXV.  Examples  of  the  rectangle  1.382  = .7236 


The  rectangle  2.382  is  found  as  the  containing  area  of  two  kylikes  without 
their  handles  (nos.  137,  139).  Two  others,  including  their  handles  (nos.  155,  156), 
are  enclosed  in  the  rectangle  3.382. 

The  ratios  .382  and  .500  (half  a square),  added  together,  make  with  unity  a 
rectangle  with  the  ratio  1.1338,  which  is  more  easily  intelligible  in  terms  of  the 
reciprocal  .882.  Two  kraters,  nos.  79,  80,  are  contained  in  this  rectangle.  Cf. 
diagram  XXVI. 


Diagram  XXVI.  Examples  of  the  rectangle 
1.1338  = .882 


The  \/5  rectangle  occurs  only  three  times  as  a containing  area  among  the 
vases  analysed  in  this  book.  These  are  the  kylikes  nos.  132,  138,  without  their 
handles,  and  the  black-figured  lekythos,  no.  163.  In  both  the  kylikes  the  diam- 

[ 12  ] 


INTRODUCTION 


eter  of  the  foot  equals  the  height  of  the  vase.  In  other  words,  it  is  exactly  con- 
tained in  the  central  square  of  the  V 5 rectangle.  Cf . diagram  XXVII. 

Two  a/5  rectangles  placed  vertically  side  by  side  constitute  an  important 
area.  Its  ratio  is  1.118  (2.236-f-2).  Its  reciprocal  is  .8944  (.4472X2).  Seven 


Diagram  XXVII.  Examples  of  the  V 5 rectangle  (2.236  = .4472) 

examples  are  illustrated  in  diagram  XXVIII.  They  are  the  stamnos,  no.  51,  the 
kalpis,  no.  67,  the  black-figured  krater,  no.  73,  the  red-figured  olpe,  no.  100,  the 
black-figured  skyphos,  no.  108,  the  red-figured  skyphos,  no.  116,  the  red-figured 
kantharos,  no.  121. 

The  bowl  of  the  black-figured  skyphos,  no.  104,  is  enclosed  in  three  \/5  rec- 
tangles placed  side  by  side  (ratio:  1.3416);  and  the  same  vase,  with  its  handles, 


is  contained  in  an  area  made  up  of  four  \/5  rectangles  (ratio:  1.7888).  The 
skyphos,  no.  107,  also  has  the  ratio  1.7888. 

The  area  made  up  of  a square  plus  a/5  rectangle,  which  has  been  found  by 
Mr.  Hambidge  to  be  the  basis  of  the  proportions  of  the  Parthenon,  occurs  at 

C 13  ] 


GEOMETRY  OF  GREEK  VASES 


least  seven  times  among  the  vases  studied  in  this  book.  Its  ratio  is  1.4472  (recip- 
rocal: .691).  The  examples  illustrated  in  diagram  XXIX  are  two  black-figured 
amphorae,  nos.  7,  8,  a red-figured  hydria,  no.  63,  a black-figured  deinos  and 


Diagram  XXIX.  Examples  of  the  rectangle  1.4472  = .691 


stand,  no.  72,  a cup  with  impressed  decoration,  no.  118,  a kantharos,  no.  123,  and 
a black-figured  kylix  with  its  handles  omitted,  no.  129. 

One  half  of  1.4472  equals  .7236.  This  is  the  reciprocal  of  1.382,  showing  that 
the  1.382  rectangle  is  composed  of  two  1.4472  rectangles.  Cf.  diagram  XXX. 
The  relation  appears  also  in  the  following  equation : 


1 

1.4472 


= .691  = 


1.382 

2 


It  is  noteworthy  that  one  of  the  two  amphorae  signed  by  Amasis,  which  are 
technically  among  the  finest  pieces  in  the  collection,  is  enclosed  in  the  rectangle 
1.382  (no.  4),  while  the  other  (no.  7)  is  contained  in  the  rectangle  1.4472. 


s 

V5 

s 

V5 

Diagram  XXX.  Relation  between 
the  rectangles  1.4472  and  1.382 


Two  1.4472  shapes,  placed  one  above  the  other,  have  the  ratio  2.8944  (recip- 
rocal: .3455  = .691  -f- 2).  Two  lekythoi,  nos.  166,  170,  exhibit  this  proportion. 
The  lekythoi  are  illustrated  in  diagram  XXXI.  The  lid  of  the  stamnos,  no.  51, 
is  also  contained  in  this  rectangle. 


[ 14  ] 


INTRODUCTION 


If  a square  is  subtracted  from  this  rectangle,  the  ratio  becomes  1.8944.  The 
reciprocal  of  this  is  .52787,  generally  given  as  .528  in  this  book.  A skyphos,  no. 
Ill,  and  two  lekythoi,  nos.  165,  179,  fit  this  rectangle.  The  lekythoi  are  illus- 
trated in  diagram  XXXI. 

Several  areas  which  are  close  to  a square  are  sometimes  difficult  to  distinguish 
from  one  another,  though  in  most  examples  a thorough  study  of  the  details 


establishes  clearly  the  ratio  of  the  over-all  rectangle.  The  shapes  1.0787  (.927), 
1.118  (.8944)  and  1.1338  (.882)  have  already  been  discussed.  A fine  red-figured 
kalyx  krater,  no.  76,  is  contained  in  the  rectangle  1.090,  or,  more  accurately, 
1.0902,  which  is  made  up  of  .618  and  .472  (better:  .4722).  A stamnos,  no.  56, 
and  a column  krater,  no.  74,  are  enclosed  in  the  rectangle  1.0225  (.618+.4045), 
the  reciprocal  of  which  is  .978.  The  analysis  of  the  stamnos  is  unusually  clear. 
The  shape  1.0557  is  more  easily  comprehended  in  terms  of  its  reciprocal,  .9472, 
which  is  made  up  of  .4472  and  .500  — a -\/5  rectangle  added  to  half  a square. 


n 

n 

7 

Diagram  XXXII.  Examples  of  the  rectangle 
1.0557  = .9472 

A stamnos,  no.  52,  and  a kantharos  signed  by  Hieron,  no.  123,  are  contained  in 
this  area.  Cf.  diagram  XXXII. 

A \/5  rectangle  (.4472)  combined  with  a whirling  square  rectangle  (.618) 
make  the  area  1.0652  (reciprocal:  .9388)  which  is  illustrated  by  the  kalpis,  no.  64, 
and  the  olpe,  no.  98.  The  red-figured  oinochoe,  no.  89,  is  enclosed  in  the  rectangle 

[ 15  ] 


GEOMETRY  OF  GREEK  VASES 


2.0652.  The  ratio  appears  also  in  the  Corinthian  skyphos,  no.  103,  and  the  black- 
figured  kylix,  no.  125.  Nos.  64,  89,  and  98  are  shown  in  diagram  XXXIII. 

The  area  1.1708,  more  easily  recognisable  in  terms  of  the  reciprocal,  .854, 
occurs  several  times,  but  the  analysis  of  this  rectangle  is  unusually  difficult.  The 
ratio  .854  equals  (.618X3)  — 1.000,  or  .618+. 236.  This  is  the  containing  area  of 
the  kantharos,  no.  120,  and  probably  of  two  or  three  other  vases  in  the  collection. 


1.0652  and  2.0652 


Two  other  shapes  which  are  close  to  a square  — 1.0355  and  1.0606  — are 
based  on  the  y/2  rectangle,  and  will  be  considered  below. 

The  area  2.472  (.618X4)  was  mentioned  above  as  the  containing  shape  of  two 
red-figured  kylikes.  If  a square  is  subtracted,  the  rectangle  1.472  (reciprocal: 
.6793),  is  obtained.  It  is  composed  of  a square  plus  two  .236  ( = 4.236)  rectangles, 
and  can  be  subdivided  in  various  ways.  Five  black-figured  amphorae,  nos.  9-13, 
a black-figured  oinochoe,  no.  85,  a Corinthian  kylix,  omitting  its  handles,  no.  103, 


Diagram  XXXIV.  Examples  of  the  rectangle  1.472  = .6793 


and  a perfume  vase  of  the  black-figured  period,  no.  184,  illustrate  this  shape. 
Cf.  diagram  XXXIV. 

Another  favorite  area  is  the  rectangle  1.528  (reciprocal:  .6545).  This  may  be 
subdivided  in  various  ways.  In  diagram  XXXV,  figure  a shows  the  area  divided 


[ 16  ] 


INTRODUCTION 


into  four  .382  ( = 2.618)  shapes.  1.528  = 2 = .764.  This  is  the  reciprocal  of  1.309: 
cf.  diagram  b.  The  two  1.309  rectangles  can  be  further  subdivided  as  shown  in 
diagram  c.  The  division  1.000 +.528  is  shown  in  diagram  d.  The  ratio  .528  is  the 
reciprocal  of  1.8944,  and  is  composed  of  a square  plus  two  s/5  rectangles.  A 
further  subdivision  is  shown  in  diagram  e.  A black-figured  amphora,  no.  15,  a 
red-figured  amphora,  no.  40,  a hydria  with  its  handles  omitted,  no.  63,  a red- 


5 

W 

w 

5 

w 

w 

5 

w 

5 

5 

w 

S 

5 

w 

5 

S 

w 

5 

W 

V5 

W 

w 

5 

W 

w 

5 

W 

w 

V5 

w 

V5 

5 

V5 

V5 

5 

V5 

S 

5 

V5 

5 

5 

V5 

S 

V5 

5 

V5 

Diagram  XXXV.  Subdivisions  of  the  rectangle  1.528  = .6545 

figured  oinochoe,  no.  95,  and  a Corinthian  skyphos,  no.  102,  are  enclosed  in  this 
area.  The  rectangle  2.528  is  illustrated  by  the  kylikes,  nos.  142,  143;  the  rec- 
tangle 3.528  by  the  kylix,  no.  157. 

The  rectangle  1.809  (reciprocal:  .5528),  made  up  of  a square  plus  half  a whirl- 
ing square  rectangle,  is  the  containing  area  of  four  Nolan  amphorae,  nos.  33-36, 
and  two  red-figured  skyphoi,  nos.  115,  117.  Cf.  diagram  XXXVI.  This  area 
with  a square  added,  i.  e.,  2.809,  encloses  two  red-figured  lekythoi,  nos.  168,  169. 

The  generating  form  of  all  the  rectangles  — the  square  — occurs  less  often 
than  might  be  expected.  Nine  examples  are  illustrated  in  diagram  XXXVII  as 
follows:  two  stamnoi,  nos.  52,  53;  a red-figured  hydria,  no.  63;  a red-figured 


Diagram  XXXVI.  Examples  of  the  rectangle  1.809  = .5528 


kalpis,  no.  64;  a black-figured  krater,  no.  73;  a red-figured  krater,  no.  77 ; a red- 
figured  oinochoe,  no.  92;  a cup  with  impressed  decoration,  no.  118;  a polychrome 
pyxis,  no.  182.  The  oinochoe,  no.  93,  is  omitted  because  it  is  a duplicate  of  no.  92. 
The  small  olpe,  no.  99,  brings  the  total  number  of  examples  up  to  eleven.  It  is 

[ 17  ] 


GEOMETRY  OF  GREEK  VASES 


probable  that  many  more  “hidden  squares”  would  be  revealed  by  further  study. 
Seven  of  the  eleven  examples  have  proportions  based  on  the  whirling  square 
rectangle  (nos.  52,  63,  64,  73,  92,  93,  118).  The  proportions  of  the  krater,  no.  77, 
and  of  the  pyxis,  no.  182,  are  in  terms  of  y/2.  The  stamnos,  no.  53,  is  an  example 
of  static  symmetry.  It  is  noteworthy  that  in  this  example  not  only  the  over-all 
rectangle,  but  also  the  rectangle  containing  the  vase  up  to  the  shoulder  and 
omitting  the  handles  is  a square. 

The  \/4  rectangle,  composed  of  two  squares  (ratio:  2.000;  reciprocal:  .500) 
occurs  three  times  as  the  containing  area  of  a complete  vase,  and  six  times  as  the 


containing  area  of  an  important  part  of  a vase.  The  examples  are:  the  kantharos, 
no.  122  (to  lip);  the  black-figured  kylikes,  nos.  128,  129;  the  black-figured 
lekythos,  no.  164;  the  red-figured  lekythoi,  nos.  168,  170,  171,  174  (to  shoulder, 
in  each  case);  the  black-figured  perfume  vase,  no.  183  (without  its  foot).  Cf. 
diagram  XXXVIII.  Nos.  128  and  164  are  examples  of  static  symmetry.  The 
others  have  proportions  based  on  the  whirling  square  rectangle.  The  ratio  .500 
occurs  very  often  among  the  proportions  of  details. 

The  \/9  rectangle,  made  up  of  three  squares,  ratio:  3.000;  reciprocal:  .333,  is 
found  seven  times  among  the  vases  studied.  The  examples  are : the  loutrophoros, 
no.  43;  the  red-figured  kylikes,  nos.  135-137;  the  red-figured  lekythoi,  nos.  172- 
174.  Cf.  diagram  XXXIX.  All  these  examples  have  the  proportions  of  their 
details  in  terms  of  the  whirling  square  rectangle. 

The  rectangle  composed  of  a square  and  a half  (ratio:  1.500;  reciprocal:  .666) 
encloses  several  black-figured  amphorae  with  apparently  “static”  proportions. 

[18] 


INTRODUCTION 


The  black-figured  kylix,  no.  128,  without  its  handles,  is  contained  in  the  same 
area;  its  proportions  are  also  static.  The  midget  pointed  amphora,  no.  44,  up  to 
its  shoulder,  is  another  example.  The  proportions  of  this  are  in  terms  of  y/2. 
Two  hydriae,  nos.  61  and  62,  without  their  handles,  are  also  enclosed  in  this  shape. 


The  red-figured  lekythos  signed  by  the  potter  Gales,  no.  167,  is  contained  in  a 
rectangle  composed  of  two  and  a half  squares  (ratio:  2.500;  reciprocal:  .400). 
Its  proportions  are  static  throughout  (cf.  above,  page  2). 

Consideration  of  the  occurrences  of  squares  and  multiples  of  squares  in  the 
vases  here  published  has  brought  us  to  the  subject  of  static  symmetry,  about 


which  little  need  be  said,  because  of  the  scanty  and  unsatisfactory  material 
offered  in  this  book.  Nine  vases  have  proportions  which  can  be  expressed  in 
simple  fractions  as  well  as  in  geometrical  constructions  (cf.  the  table  below,  page 
27).  The  black-figured  amphorae  nos.  23-25,  on  the  other  hand,  though  their 

[ 19  ] 


GEOMETRY  OF  GREEK  VASES 


diameter  is  to  their  height  as  1 is  to  1.5,  have  proportions  which  I have  been  un- 
able to  express  intelligibly  either  in  areas  or  in  linear  units.  Certain  “static” 
ratios,  especially  .500,  and,  less  frequently,  .333,  1.000,  .750,  .666,  .250,  occur  in 
vases  belonging  to  the  “dynamic”  class.  It  remains  to  examine  the  vases  with 
proportions  based  on  the  y/2  and  \/3  rectangles. 

The  \/2  rectangle  (ratio:  1.4142;  reciprocal:  .7071)  occurs  six  times  as  a 
containing  area,  the  examples  being  a black-figured  amphora,  no.  22;  a red- 
figured  pelike,  no.  50;  two  red-figured  oinochoai,  nos.  90,  96;  a black-figured 


Diagram  XL.  Examples  of  the  V 2 rectangle  (1.4142  = .7071) 

skyphos,  no.  105,  omitting  its  handles.  The  black-figured  hydria,  no.  57,  up  to 
its  shoulder,  is  also  contained  in  a y/2  rectangle.  Cf.  diagram  XL. 

Eight  areas  derived  from  the  \/2  rectangle  are  of  sufficient  interest  to  be 
mentioned  here. 

Three  y/2  rectangles  placed  vertically  side  by  side  compose  an  area  with  the 
ratio  2.1213  (reciprocal:  .4714).  A fine  black-figured  kylix  signed  by  Tleson,  no. 
124,  and  the  lid  of  the  polychrome  pyxis,  no.  182,  illustrate  this  shape.  Cf. 
diagram  XLI. 

If  this  rectangle  is  bisected  vertically  each  half  has  the  ratio  1.0606  (recip- 
rocal: .9428),  and  is  composed  of  sixv/2  rectangles  placed  in  two  rows  of  three 
each.  This  is  the  area  in  which  the  black-figured  hydria,  no.  57,  is  enclosed,  if  the 
projection  of  the  vertical  handle  above  the  lip  is  omitted.  Cf.  the  small  diagram  C 
on  page  103. 

If  two  squares  are  added,  the  ratio  just  mentioned  is  changed  to  3.0606.  This 
is  the  rectangle  enclosing  the  kylix  decorated  by  Oltos,  no.  134. 

[ 20  ] 


INTRODUCTION 


A square  with  a y/2  rectangle  added  has  the  ratio  1.7071  (reciprocal:  .5858). 
The  red-figured  lekythos,  no.  178,  is  enclosed  in  this  area. 

Two  squares  plus  a y/2  rectangle  make  an  area  with  the  ratio  2.7071  (recip- 
rocal: .3693).  Three  red-figured  kylikes  without  their  handles,  nos.  151,  152,  154, 
are  placed  in  this  rectangle. 

One  of  the  most  interesting  areas  based  on  the  -y/2  rectangle  is  the  rectangle 
1.2071  (reciprocal : .8284),  which  is  composed  of  a y/2  rectangle  plus  half  a square 
(.7071 + .500).  The  black-figured  amphora,  no.  2,  the  black-figured  hydria,  no. 
57,  with  its  lateral  handles  omitted,  and  the  polychrome  pyxis,  no.  182,  are  con- 


tained in  this  area.  Cf.  diagram  XLII.  (Cf.  also  the  pelike  no.  45  and  the  hydria 
no  61.) 

This  shape  with  a square  added  (ratio:  2.2071;  reciprocal:  .4531)  encloses  the 
small,  pointed  amphora,  no.  44. 

The  ratio  .2071,  which  is  the  reciprocal  of  4.8284,  represents  an  area  made  up 
of  two  squares  and  two  y/2  rectangles.  If  five  of  these  areas  are  placed  vertically 
side  by  side,  the  resulting  rectangle  has  the  ratio  1.0355  (.2071 X 5),  the  reciprocal 
of  which  is  .9657.  Cf.  diagram  XLIII.  The  black-figured  hydria,  no.  57,  is 
contained  in  this  shape. 

A rectangle  made  up  of  a y/2  rectangle  and  two  squares  (ratio:  3.4142;  recip- 
rocal: .2929)  contains  the  kylikes,  nos.  148-152,  with  their  handles  included. 
Without  their  handles  most  of  these  kylikes  have  the  ratio  2.7071,  mentioned 
above.  One  of  them,  no.  148,  has  the  ratio  2.4714,  which  can  be  obtained  by 
subtracting  .9428,  the  reciprocal  of  1.0606  from  3.4142.  The  ratio  .4714  is  the 
reciprocal  of  2.1213,  or  three  y/2  rectangles  placed  vertically  side  by  side.  Cf. 
diagram  XLI. 


[ *1.] 


GEOMETRY  OF  GREEK  VASES 


s 

s 

s 

s 

a/2 

V2 

V2 

a/2 

Diagram  XLIII. 

The  rectangle  1.0355  = .9657 


The  a/ 3 rectangle  occurs  as  the  containing  shape  of  three  Nolan  amphorae, 
nos.  29-31. 

Another  red-figured  amphora,  no.  43,  is  enclosed  in  an  area  composed  of  a 
square  with  a \/3  rectangle  added  to  one  side.  Ratio  1.5773. 

A black-figured  kylix  signed  by  Xenokles,  no.  126,  is  apparently  contained  in 
an  area  based  on  the  \/3  rectangle.  The  ratio  of  width  to  height  is  2.0206,  or 

V3+^,  i.  e.,  1. 732+. 2886. 

6 

In  the  foregoing  pages  fifty-four  areas  have  been  defined  which  occur  more  than 
two  hundred  times  among  the  vases  analysed  in  this  book.  Speaking  in  general 
terms,  each  area  is  found,  on  an  average,  four  times.  Many  of  the  vases,  however, 
illustrate  two,  three,  four,  or  even  five  different  rectangles,  according  as  the 
whole  or  some  important  part  is  considered.  A striking  instance  of  this  is  fur- 
nished by  the  stamnos,  no.  51,  which  exhibits  five  familiar  areas: 


The  complete  stamnos  with  its  lid 1.118  = .8944 

The  complete  stamnos  without  its  lid 1.0787  = .927 

The  stamnos  up  to  the  shoulder 1.236  =.809 

The  stamnos  without  lid  and  handles 1.191  = .8396 

The  lid  considered  separately 2.8944  = .3455 


The  number  of  different  vases  which  illustrate  these  fifty-four  areas  is  con- 
sequently only  one  hundred  and  thirty-four.  Since  one  hundred  and  eighty-five 
vases  are  published,  it  is  obvious  that  the  total  number  of  areas  must  be  far 
greater  than  fifty-four.  As  a matter  of  fact  the  number  is  at  least  ninety. 
Judging  the  problem  merely  on  the  basis  of  these  statistics,  one  might  be  justified 
in  inferring  that  the  shapes  described  are  devoid  of  significance.  Every  object 
under  the  sun  can  be  enclosed  in  a rectangle;  and,  if  a sufficient  number  of  objects 
are  measured,  recurrences  of  some  of  the  rectangles  are  bound  to  appear.  In  the 
same  way  it  might  be  argued  that  the  ratios  which  describe  the  proportions  of 
details  of  the  vases  with  reference  to  the  complete  height  or  width  are  so  numerous 
that  every  detail  must  fit  approximately  some  one  of  the  ratios.  But  that  the 

[ 22  ] 


INTRODUCTION 


matter  cannot  be  dismissed  so  lightly  will  be  apparent  to  any  one  who  seriously 
examines  the  geometrical  analyses  of  the  vases  and  the  tables  of  ratios  which 
accompany  them.  Consideration  of  the  following  points  may  perhaps  facilitate 
the  study  of  the  evidence. 

1.  The  accuracy  with  which  the  vases  conform  to  the  rectangles  used  in  ana- 
lysing them  is  a question  of  some  importance.  Since  the  main  dimensions  are 
given  in  every  case  the  margin  of  error  can  easily  be  calculated.  But  to  spare  the 
reader  this  labor  the  results  of  such  a calculation  are  presented  here  in  tabular 
form.  In  each  case  I have  multiplied  the  smaller  dimension  by  the  ratio,  thus 
getting  the  maxim  error.  Multiplying  the  larger  dimension  by  the  reciprocal 
would  have  decreased  the  error;  and  by  distributing  the  error  between  height  and 
width,  it  could  have  been  still  farther  reduced.  The  total  number  of  proportions 
thus  investigated  is  277.  Many  of  the  vases  furnished  more  than  one  rectangle. 
It  seemed  worth  while,  in  vases  like  the  stamnos,  hydria,  oinochoe,  skyphos, 
kantharos,  and  kylix  to  examine  the  proportions  with  and  without  the  handles, 
and  where  a vase  had  a lid  to  consider  it  with  and  without  the  lid.  Fifteen  ex- 
amples were  eliminated  either  because  restorations  made  them  unreliable,  or  be- 
cause they  did  not  fit  any  recognisable  proportions.  The  remaining  263  examples 
were  grouped  as  follows: 

A Error  indistinguishable. 

B Error  less  than  one  millimetre. 

C Error  between  one  and  two  millimetres. 

D Error  between  two  and  three  millimetres. 

E Error  between  three  and  four  millimetres. 

F Error  greater  than  four  millimetres. 

The  results  of  the  calculation  may  be  summarized  as  follows: 

A-C  Error  less  than  two  millimetres.  195  examples. 

D-F  Error  greater  than  two  millimetres.  68  examples. 

The  examples  in  the  second  group  require  further  scrutiny.  Of  the  fifteen 
amphorae  included  four  could  not  be  analysed  (nos.  6,  21,  24,  25),  and  in  four 
other  cases  (nos.  16,  19,  22,  31)  the  analysis  is  to  be  regarded  as  merely  tentative. 
The  stamnos  no.  54  is  to  be  eliminated  as  a failure.  The  krater  no.  77,  is  a doubt- 
ful example.  Nos.  73  and  79,  on  the  other  hand,  are  admitted  because  the 
correctness  of  the  schemes  adopted  was  strikingly  confirmed  by  the  details.  The 
red-figured  kylikes  are  a special  case.  The  widely  spreading  bowls  are  often  not  set 
quite  horizontally  upon  the  slender  stems,  so  that  the  intended  height  is  difficult 
to  determine.  A variation  of  one  millimetre  in  the  height  changes  the  width  and 
the  diameter  of  the  bowl  about  three  millimetres.  In  the  analyses  a theoretical 
height  is  usually  taken;  and  this  is  carefully  noted  in  every  case.  Forty-two  ex- 
amples may,  therefore,  fairly  be  removed  from  the  second  group,  which  is  thus 
reduced  to  twenty-six  examples.  In  general  the  maximum  error  admitted  is 
under  two  millimetres.  I have  no  figures  to  show  the  accuracy  of  details,  but  I am, 

[ 23  ] 


GEOMETRY  OF  GREEK  VASES 


confident  that  in  the  very  large  majority  of  cases  the  error  is  well  under  two  milli- 
metres. At  first  glance  this  margin  of  inaccuracy  perhaps  appears  too  small. 
When  one  takes  into  account  the  possible  carelessness  of  so  humble  an  artisan  as 
a potter  in  conforming  to  a given  scheme,  the  difficulty  of  accurately  assembling 
the  various  parts  of  a vase  before  the  clay  has  hardened,  the  shrinkage  in  firing, 


Number 

of 

examples 

Amount  of  Error 

A 

B 

c 

D 

E 

F 

Amphora 

40 

4 

8 

13 

6 

3 

6 

Pelike 

6 

2 

1 

3 

Stamnos 

6 

1 

3 

1 

1 

Stamnos  details 

7 

1 

5 

1 

Hydria 

12 

2 

7 

2 

1 

Hydria  details 

11 

1 

6 

1 

3 

Deinos 

3 

1 

1 

1 

Deinos  and  stand 

1 

1 

Demos,  stand  alone 

1 

1 

Krater 

10 

2 

3 

2 

2 

1 

Krater  details 

2 

1 

1 

Psykter 

1 

1 

Oinbchoe 

16 

4 

9 

3 

Oinochoe  without  handle 

8 

3 

5 

Skyphos 

15 

3 

8 

2 

2 

Skyphos  without  handles 

15 

10 

2 

3 

Kantharos 

4 

1 

3 

Kantharos  without  handles 

3 

1 

1 

1 

Kylix 

37 

3 

9 

7 

6 

6 

6 

Kylix  without  handles 

38 

3 

14 

7 

6 

8 

Leky thos 

17 

2 

9 

4 

1 

1 

Pyxis 

2 

2 

Pyxis  without  lid 

2 

2 

Perfume  vase 

3 

3 

Perfume  vase  without  lid 

3 

3 

Total 

263 

33 

112 

50 

31 

23 

14 

one  is  inclined  to  admit  a more  considerable  error.  And  by  doing  so  the  confus- 
ingly large  number  of  ratios  could  very  likely  be  reduced.  On  the  other  hand,  any 
one  who  has  carefully  examined  masterpieces  of  pottery  like  the  Brygan  kan- 
tharos,  no.  121,  will  agree  that  such  a vase  must  have  been  fashioned  with  almost 
as  much  care  as  the  vases  of  precious  metal  whose  forms  it  imitates.  It  was  not 
merely  “thrown”  on  the  wheel,  but  carefully  “turned”  after  the  clay  had  be- 
come “leather  hard.”  The  clay  is  of  extremely  fine  texture,  the  walls  astonish- 
ingly thin;  and  it  is  a question  whether  the  firing  would  appreciably  affect  the 

[ 24  ] 


INTRODUCTION 


proportions,  though  it  would  decrease  the  dimensions.  All  things  considered,  it  is 
safer  to  follow  the  evidence  we  possess,  rather  than  to  juggle  with  it  in  the  hope  of 
achieving  more  striking  results. 

2.  As  was  remarked  above  there  is  no  significance  in  the  fact  that  Greek  vases 
can  be  contained  in  the  rectangles  of  dynamic  symmetry.  It  may,  however,  be 
significant  that  (1)  a large  proportion  of  the  vases  conform  accurately  to  a limited 
number  of  comparatively  simple  rectangles,  that  (2)  in  a large  proportion  of  ex- 
amples all  the  details  can  be  accurately  expressed  in  terms  of  the  containing  rec- 


Ratio 

No.  of 
occur- 
rences 

A 

B 

1.236  = .809 

21 

46,  51,  55,  56,  73,  91,  95, 107, 109, 
110, 111, 112, 113, 115, 120, 185 

18, 19,  73,  76,80 

1.000 

15 

53,  55,  73,  77,99,  118 

16,52,  53, 63,64,92,93,182 

1.618  = .618 

12 

9, 17, 18,  41,  69,  85,  86,  87,  92,  93, 
97, 185 

1.118  = .8944 

11 

51,  67,  68,  73, 100, 108, 116, 121, 121 

9, 17 

1.382  = .7236 

10 

4, 5, 49, 67, 69, 104, 123, 130, 131, 183 

2.000  = .500 

9 

122, 128, 129, 164 

168, 170, 171, 174, 183 

1.309  = .764 

8 

47,  65,68,92,93, 117 

35,41 

1.4472  = .691 

8 

6,7,8,  63,72,118,123, 129 

3,236  = .309 

8 

133, 141, 142, 143, 144, 145, 146, 147 

1.472  = .6793 

7 

9,10,11,12, 13,85, 103 

1.500  = .666 

7 

14,21,23,24,44,61,62 

1,854  = .5394 

7 

37, 88, 104, 108, 109, 171, 183 

3.000  = .333 

7 

43, 135, 136,  137, 172, 173, 174 

1.809  = .5528 

6 

33,  34,35,  36,115,117 

2.618  = .382 

6 

133, 145, 146, 147, 155, 181 

1.0787  = .927 

5 

51,  55,  58,  65,  66 

1.2071  = .8284 

5 

2,45,57,61, 182 

1.4142  = .7071 

5 

20, 22, 90, 96, 105 

1.528  = .6545 

5 

18,  39,  63,  95, 102 

3.4142  = .2929 

5 

148,149,150,151,153 

20  Ratios 

167 

tangles,  and  that  (3)  a large  majority  of  the  details  coincide  accurately  with  a 
small  number  of  simple  ratios.  In  connection  with  the  first  point  it  seems  worth 
while  to  give  a full  list  of  the  occurrences  of  the  twenty  most  common  containing 
areas,  classifying  them  according  as  they  contain  (A)  whole  vases  with  or  without 
their  lids  and  handles,  and  (B)  important  parts  of  vases. 

The  significance  of  the  second  point  can  only  be  judged  by  studying  the  draw- 
ings with  the  accompanying  explanations.  As  regards  the  third  point  the  follow- 
ing statistics  are  submitted.  In  compiling  them  I have  limited  myself  to  the 
vases  analysed  in  terms  of  V 5.  These  furnished  99  ratios  less  than  unity,  which 

C 25  ] 


GEOMETRY  OF  GREEK  VASES 


occurred  in  all  591  times.  The  thirty  most  popular  ratios  are  given  here  with  the 
number  of  occurrences  of  each: 


Ratios 

Occurrences 

Ratios 

Occurrences 

Ratios 

Occurrences 

.382 

52 

.5669 

14 

.7888 

7 

.500 

46 

.5528 

13 

.191 

7 

.309 

40 

.708 

13 

.354 

7 - 

.4472 

32 

.4045 

13 

.118 

6 

.236 

32 

.6584 

11 

.2236 

6 

.618 

30 

.250 

11 

.333 

6 

.764 

21 

.472 

10 

.1459 

5 

.528 

19 

.854 

9 

.264 

5 

.691 

18 

.7236 

8 

.5394 

5 

.809 

15 

.2764 

8 

.1708 

4 

10 

305 

10 

110 

10 

58 

The  ten  ratios  in  the  first  column  occur  more  frequently  than  the  remaining 
eighty-nine  ratios.  And  the  thirty  ratios  here  listed  occur  473  times,  whereas  the 
remaining  sixty-nine  ratios  occur  only  118  times. 

Of  the  twenty-nine  intervals  between  these  ratios  only  nine  are  small  enough 
to  cause  confusion  in  fixing  the  ratio  in  an  example  whose  smaller  dhnension  is 
0.20  m.  In  actual  practice  I have  found  that  doubt  as  to  the  choice  between  two 
ratios  very  rarely  arises.  In  twenty-eight  amphorae,  for  example,  whose  average 
diameter  was  24  cm.,  the  difference  between  thirty- three  pairs  of  adjacent  ratios 
was  over  3 mm.  in  twenty-seven  cases. 

3.  The  following  table  gives  a classification  of  the  vases  according  to  the  sys- 
tem of  analysis  used.  The  examples  which  worked  out  successfully  are  marked  A; 
those  which  worked  out  fairly  well  are  placed  in  column  B ; and  those  which  were 
regarded  as  unsatisfactory  in  column  C. 

4.  In  a criticism  of  Dynamic  Symmetry,  published  in  the  American  Journal  of 
Archaeology , XXV,  1921,  pp.  18  ff.,  Professor  Carpenter  has  raised  the  question 
whether  the  analyses  in  Mr.  Hambidge’s  book  are  not  for  the  most  part  “mere 
adroit  manipulation,  combined  with  a mystifying  conversion  of  very  simple 
linear  ratios  into  a guise  of  root-rectangles.  ” To  illustrate  his  point  he  places  side 
by  side  Mr.  Hambidge’s  “dynamic”  analysis  of  a lekythos  at  Yale  and  his  own 
static  analysis  of  the  same  vase.  He  justly  remarks  that  the  issue  here  involved  is 
crucial.  He  also  notes  that  many  of  the  most  frequent  and  important  “effective 
numbers”  happen  to  fall  very  close  to  certain  simple  “static”  ratios,  as  follows: 


2.236  (V  5)  is  scarcely  distinguishable  from 2.25  =9:4 

2.000  is  identical  with 2 =8:4  = 2: 1 

1.732  is  scarcely  distinguishable  from 1.75  =7:4 

1.618  (the  redoubtable  "whirling  square  ” ratio)  and  . . 1.6  =8:5 

.618  (its  reciprocal)  closely  approximate 625  = 5:8 

1.309  agrees  very  nearly  with 1.333  = 4:3 


[ 26  ] 


INTRODUCTION 


C 27  ] 


GEOMETRY  OF  GREEK  VASES 


He  finds  that  “here  more  than  anywhere  else  lies  the  key  to  Mr.  Hambidge’s 
ingenious  magic,”  and  reminds  us  that  “a  ratio  approximating  5:  8 has  in  all  ages 
been  a recurring  favorite  in  artistic  composition  and  artistic  design.  . . . Some- 
where in  the  neighborhood  of  that  ratio,  man  has  an  inveterate  tendency  to 
localise  his  sense  for  beauty  of  proportions.  For  the  old  potter  working  with  a 
simple  rule,  that  ratio  was  a natural  one  to  employ.  Continued  bisection  of  his 
rule  would  give  him  8 parts  or  16  parts  with  which  to  lay  out  and  measure.  It  was 
only  to  be  expected  that  he  should  often  avail  himself  of  that  harmonious  division 
into  a little  more  and  a little  less  than  half  which  f or  H would  give  him.  Wherever 
he  used  this  ratio,  the  dynamic  analyst  will  be  able  to  discover  ‘ whirling  squares/ 
since  f is  a remarkably  close  approximation  to  the  division  into  extreme  and  mean 
proportion  from  which  the  'whirling  square’  rectangle  derives  its  peculiar  prop- 
erties of  subdivision.” 

In  short  Mr.  Carpenter  prefers  to  return  to  the  Vitruvian  method  of  studying 
Greek  proportions,  which,  as  regards  architecture,  had  long  since  been  abandoned 
as  a hopeless  failure,  and,  for  that  reason  perhaps,  has  never  been  seriously  ap- 
plied to  Attic  pottery.  His  remarks  call  for  consideration  here,  if  only  to  make 
clear  the  purpose  of  this  book.  It  is  not  published  as  an  argument  for  or  against 
the  theory  that  the  Attic  potters  consciously  used  the  systems  of  proportion  dis- 
covered by  Mr.  Hainbidge,  nor  as  an  argument  for  or  against  the  theory  that  a 
work  of  art  designed  according  to  these  systems  is  “better”  than  one  designed 
according  to  another  system,  or  according  to  no  system  at  all.  Its  aim  is  to  present 
in  as  complete  and  accurate  and  intelligible  a form  as  possible  the  evidence  fur- 
nished by  the  whole  collection  of  Attic  pottery  in  the  Museum  of  Fine  Arts. 
Many  pieces  had  to  be  rejected  because  they  were  incomplete,  some  others  because 
they  were  badly  made,  and  consequently  could  not  be  accurately  measured.  A 
considerable  number  of  lekythoi  were  omitted  because  this  type  of  vase  is  not  a 
satisfactory  subject  for  analysis.  Most  red-figured  lekythoi  are  nearly  three  times 
as  high  as  they  are  wide;  a small  change  in  the  diameter  produces  a three-fold 
change  in  the  ratio  of  the  height.  And  the  details,  except  for  the  height  of  the 
body,  are  all  at  a very  small  scale.  A few  black-figured  amphorae  and  a dozen  red- 
figured  kylikes  have  been  excluded  because  these  classes  of  vases  are  represented 
in  the  collection  in  much  greater  numbers  than  any  other  shapes.  The  pieces 
omitted  are  those  which  could  not  be  satisfactorily  analysed.  With  these  excep- 
tions, however,  the  vases  are  not  select  pieces,  but  a representative  collection  of 
admittedly  high  quality.  The  reader  is  left  to  draw  his  own  conclusions  as 
regards  the  probability  or  improbability  of  Mr.  Hambidge’s  theory.  Sufficient 
measurements  are  given  to  enable  him  to  test  also  the  validity  of  the  theory  of 
Vitruvius  and  of  Mr.  Carpenter.  My  own  experiments  along  this  line  have  not 
been  successful.  It  is  true  that  the  proportions  of  a certain  number  of  vases  can  be 
expressed  in  simple  linear  units,  but  in  every  case  one  or  more  or  all  of  the  follow- 
ing obstacles  are  encountered:  (1)  The  unit  chosen  must  be  arbitrary,  not  some 
simple  division  of  the  Greek  foot.  (2)  The  unit  must  be  made  very  small,  so  that 
the  proportions  have  little  more  significance  than  a mere  record  of  the  dimensions 

[ 28  ] 


INTRODUCTION 


would  have.  (3)  A large  margin  of  error  must  be  admitted.  (4)  Even  if  the  pro- 
portions can  be  expressed  in  fairly  large  divisions  of  the  Greek  foot  no  reason 
appears  why  those  particular  lengths  were  chosen  rather  than  others.  To  illustrate 
these  points  let  us  examine  some  of  the  examples  cited  by  Mr.  Carpenter.  The 
proportions  of  the  lekythos  (l.c.  p.  33,  fig.  7)  seemingly  work  out  as  well  in  terms 
of  units  0.013  m.  long  as  they  do  in  terms  of  the  y/2  rectangle.  And  I agree  that 
“on  the  reader’s  judgment  of  the  issue  here  involved  will  hang  his  whole  faith 
in,  or  distrust  for,  dynamic  symmetry.”  But  an  issue  of  such  importance  cannot 
be  decided  on  the  evidence  of  one  example.  The  skyphos  in  New  York,  to  which 
he  refers  (l.c.  p.  35,  note  1)  furnishes  a better  example,  because  there  is  another 


skyphos  with  identically  the  same  proportions  : 

in  Boston  (no.  104)  and  because 

the  height  of  each  is  very  close  to  half  of  the  Aeginetan-Attic  foot.  The  dimen- 

sions  of  the  skyphoi  are  as  follows: 
Height 

New  York 

0.164  m. 

Boston 

0.160  m. 

Width 

0.304  m. 

0.298  m. 

Largest  diameter  of  bowl 

0.2255  m. 

0.221  m. 

Smallest  diameter  of  bowl 

0.100  m. 

0.099  m. 

Diameter  of  foot 

0.137  m. 

0.134  m. 

The  static  analysis  is  as  follows: 

Units 
(1  dactyl) 

New  York 

Error 

Boston 

Height 

8 

0 

0 

Width 

15 

— 0.0035  m. 

-0.002  m. 

Greatest  diameter  of  bowl 

11 

0 

— 0.001  m. 

Smallest  diameter  of  bowl 

5 

-0.0025  m. 

+0.001  m. 

Diameter  of  foot 

7 

-0.0065  m. 

— 0.006  m. 

The  dynamic  analysis  is  as  follows: 

Ratios 

Error 

Height 

1.000 

New  York 

0 

Boston 

0 

Width . 

1.854 

— 0.00006  m. 

-0.00136  m. 

Greatest  diameter  of  bowl 

1.382 

-0.00115  m. 

-0.00012  m. 

Smallest  diameter  of  bowl 

.618 

-0.00135  m. 

+0.00012  m. 

Diameter  of  foot 

.854 

— 0.003  m. 

— 0.00264  m. 

The  static  ratios  are  seen  to  be  less  accurate  than  the  dynamic,  especially  that 
of  the  diameter  of  the  foot.  The  static  analysis  shows  plainly  that  the  smallest 
diameter  of  the  bowl  is  one-third  of  the  total  width,  and  that  it  is  to  the  height  in 
the  proportion  of  5:8  (=  1.600).  Mr.  Carpenter  notes  also  that  the  lower  diam- 
eter plus  the  projection  of  the  bowl  equals  the  total  height,  and  suggests  that  this 
would  seem  to  have  been  a common  potter’s  formula,  just  as  in  speaking  cf  an- 
other skyphos  (l.c.  p.  30;  fig.  5;  no.  117  in  this  book)  he  observes  that  “the  maxi- 
mum width  is  equivalent  to  width  of  base  plus  height  of  vase,  and  the  width  of 
the  bowl  is  equivalent  to  width  of  base  plus  half  the  height  of  vase,  — which  looks 

[ 29  ] 


GEOMETRY  OF  GREEK  VASES 

like  a convenient  potter’s  formula.”  One  wonders  how  many  “ convenient” 
formulae  like  these  the  Attic  potter  kept  in  his  memory. 

The  dynamic  analysis  shows  clearly  the  same  pair  of  proportions  1:3  and  5:8; 
but  in  the  latter  case  the  exact  ratio  is  substituted  for  the  approximation.  It  re- 
veals also  other  instances  of  extreme  and  mean  proportion,  which  are  not  at  once 
apparent  from  the  list  of  static  units.  For  example,  the  diameter  of  the  foot  is  to 
the  greatest  diameter  of  the  bowl  as  1 is  to  1.618.  In  the  static  analysis  this  ap- 
pears as  a new,  but  not  very  close  approximation,  7:11  (=  1.5714).  The  height 
is  to  the  greatest  diameter  of  the  bowl  plus  the  projection  of  one  handle  as  1 is  to 

I. 618.  To  express  it  geometrically,  the  whole  vase  is  contained  in  two  overlapping 
whirling  square  rectangles,  and  the  bowl  is  contained  in  the  overlapping  portion 
(cf.  the  diagram  on  page  150).  In  the  static  analysis  this  appears  as  still  another 
approximation,  8 :13  ( = 1.625).  The  smallest  diameter  of  the  bowl  is  to  its  largest 
diameter  as  1 is  to  \/h.  This  appears  in  the  static  analysis  in  the  form  5:11  (1.200) 
instead  of  4:9  (1.250).  The  relation  of  height  to  largest  diameter  of  bowl  is  the 
same  as  that  of  smallest  diameter  of  bowl  to  diameter  of  foot  (1.382).  In  the  static 
analysis  this  appears  as  8:11  (1.375),  and  as  5:7  (1.400).  To  sum  up,  all  the  ele- 
ments of  the  skyphos  are  related  to  one  another  in  extreme  and  mean  proportion 
and  other  proportions  intimately  connected  with  it,  which  could  be  arrived  at 
with  the  greatest  ease  by  enclosing  the  preliminary  sketch  of  the  vase  in  a 1.854 
rectangle,  and  drawing  the  simplest  subdivisions  of  one  of  the  three  whirling 
square  rectangles  of  which  it  is  composed.  It  is  difficult  to  see  how  a potter  could 
have  created  so  perfectly  coordinated  a design  with  the  help  of  the  numbers  8,  15, 

II, 5,7. 

The  proportions  of  this  skyphos  could  be  expressed  with  a fair  amount  of  ac- 
curacy in  terms  of  the  Greek  foot.  This  is,  however,  by  no  means  always  the 
case.  Let  us  take  as  an  example  the  Brygan  kantharos,  no.  121,  which  is  perhaps 
the  finest  piece  of  Attic  pottery  in  the  collection.  Mr.  Carpenter  remarks  (l.c.  p. 
34,  note  1)  that  “the  whole  kantharos  can  be  constructed  “statically”  on  a meas- 
ure divided  into  eight  parts,”  and  in  response  to  an  inquiry  he  has  explained  his 
proposed  analysis  in  detail.  The  vase  is  in  excellent  preservation,  except  for  one 
handle  which  has  been  broken  and  has  a small  piece  missing  at  the  top  restored  in 
plaster.  The  other  handle  is  intact.  The  dimensions  are: 


Height  of  unbroken  handle 0.241  m. 

(Height  of  restored  handle 0.246  m.) 

Height  of  lip 0.1675  m. 

Height  of  stem 0.075  m. 

Width 0.270  m. 

Largest  diameter  of  bowl 0.1885  m. 

Diameter  of  lower  member  of  bowl 0.121m. 

Diameter  of  top  of  stem 0.031  m. 

Diameter  of  foot 0.0985  m. 

Width  of  handles ±0.0275  m. 


[ 30  ] 


INTRODUCTION 


The  height  of  the  handles  is  assumed  to  be  0.242  m.,  which  is  twice  the  diam- 
eter of  the  lower  member  of  the  bowl.  The  ratios,  with  the  amount  of  error  are  as 


follows: 

Height  to  top  of  handles 

Height  to  lip 

Projection  of  handles  above  lip.  . . . 

Height  of  stem 

Height  of  bowl  alone 

Width 

Diameter  of  lip 

Diameter  of  lower  member  of  bowl 

Diameter  of  top  of  stem 

Diameter  of  foot 

Width  of  handles 


Ratios 

Error 

1.000 

-0.001  m. 

.691 

-0.00028  m. 

.309 

— 0.0003  m. 

.309 

+0.0002  m. 

.382 

+0.00005  m. 

1.118 

— 0.0005  m. 

.7725 

+0.0016  m. 

.500 

0 

.118 

+0.0024  m. 

.4045 

+0.0006  m. 

.118 

— 0.001  m. 

The  unit  in  Mr.  Carpenter’s  analysis  is  one-eighth  of  the  diameter  of  the  lip, 
0.02356  m.,  which  has  no  recognisable  relation  to  the  Greek  foot.  His  analysis  is 
as  follows : 


Units 

Ratios 

Error 

Height  to  top  of  handles 

. . . . 10 

1.000 

0.0054  m. 

Height  to  lip 

. . . . 7 

.700 

0.00358  m. 

Projection  of  handles  above  lip 

...  3 

.300 

0.00282  m. 

Height  of  stem 

. . . . 3 

.300 

0.00432  m. 

Height  of  bowl  alone . . . 

. . . . 4 

.400 

0.00024  m. 

Width 

. . . . 12 

1.200 

0.01272  m. 

Diameter  of  lip • 

, . . . 8 

.800 

0 

Diameter  of  lower  member  of  bowl 

...  5 

.500 

0.0037  m. 

Diameter  of  foot 

..  ..  4 

.400 

0.00426  m. 

The  first  thing  that  strikes  one  in  studying  this  scheme  of  proportions  is  that 
the  vase  does  not  fit  the  scheme.  The  height  is  half  a centimetre  too  small,  the 
width  more  than  a centimetre  too  great.  Mr.  Carpenter  accounts  for  this  by 
assuming  that  the  handles  have  been  wrongly  adjusted,  and  that  the  moulding  at 
the  junction  of  bowl  and  stem  was  not  included  in  the  scheme,  but  was  added 
when  the  two  elements  were  joined.  We  have  noted  in  the  case  of  the  kantharos 
no.  119  that  the  weight  of  the  handles  pulled  the  bowl  out  of  shape,  so  that  it  is 
oval  rather  than  circular.  In  the  present  example  there  is  nothing  to  suggest  any 
such  disturbance  of  the  scheme;  the  bowl  is  perfectly  circular.  Moreover  the  list  of 
units  does  not  reveal  a logical  and  consistent  '‘theme.”  Five  is  half  of  10;  4 is  half 
of  8,  and  one- third  of  12;  3 is  one-quarter  of  12.  Extreme  and  mean  proportion 
apparently  appears  in  the  ratio  of  the  diameter  of  the  lower  member  of  the  bowl  to 
the  diameter  of  its  lip.  But  actually  this  ratio  is  1.545,  not  1.618.  The  unit,  as 
already  remarked,  bears  no  relation  to  the  Greek  foot.  The  height  of  the  vase  is 
actually  about  half  a centimetre  less  than  twelve  dactyls,  and  the  discrepancy 
can  be  explained  as  due  to  shrinkage  in  firing.  But  I have  been  unable  to  find  a 
plausible  static  scheme  using  the  dactyl  as  a unit. 

C 31  ] 


GEOMETRY  OF  GREEK  VASES 


Mr.  Hambidge’s  analysis,  on  the  other  hand,  shows  that  every  element  of  the 
vase  conforms  accurately  to  a logical  and  consistent  “theme.”  This  has  been  made 
sufficiently  clear  in  his  book  and  in  the  drawing  and  table  of  ratios  published 
below,  page  162.  It  appears  still  more  clearly  in  the  four  diagrams  here  repro- 
duced. The  first  shows  that  the  containing  area  is  composed  of  two  V 5 rectangles. 
Each  of  these  may  be  regarded  as  two  whirling  square  rectangles  overlapping 
to  the  extent  of  a square.  The  diameter  of  the  lower  member  of  the  bowl  equals 
the  side  of  this  square,  which  is  half  of  the  total  height.  The  diagonals  in  the 
second  diagram  show  that  the  main  proportions  of  the  kantharos  are  the  same 
whether  the  handles  are  included  or  omitted.  This  is  a relationship  worth  observ- 
ing, and  it  would  hardly  appear  from  a study  of  Mr.  Carpenter’s  units.  Attic  pot- 
ters do  not  seem  to  have  used  it  frequently;  at  least  only  two  other  instances  have 
been  noted  in  this  book  (nos.  53  and  79).  The  third  diagram  shows  that  if  whirling 
square  rectangles  are  applied  at  the  top  and  bottom  of  the  containing  area  the 
bowl  is  contained  in  the  overlapping  portion.  In  the  fourth  diagram  the  kantharos 
has  been  revolved  90°.  If  whirling  square  rectangles  are  applied  at  each  side,  the 
handles  and  the  top  of  the  stem  are  seen  to  be  contained  in  the  overlapping  por- 
tion. It  is  also  noteworthy  that  if  this  overlapping  portion  were  removed  the  con- 
taining rectangle  would  become  a square.  The  proportions  of  the  kantharos  can 
all  be  simply  and  accurately  expressed  in  terms  of  extreme  and  mean  proportion, 
i.  e.,  the  ratio  1.618,  or  the  whirling  square  rectangle.  This  example  alone  does 
not  prove  that  Greek  potters  consciously  employed  such  a scheme,  just  as  the 
lekythos  studied  by  Mr.  Carpenter  does  not  prove  the  contrary.  But  it  does 
seem  to  the  writer  to  furnish  stronger  evidence  in  favor  of  the  theory  than  his  ex- 
ample furnishes  against  it,  and  therefore  to  justify  a geometrical  investigation  of 
the  proportions  of  Attic  pottery.  It  is  unjust  to  characterise  such  a procedure  as 
“ingenious  magic,”  “adroit  manipulation”  and  “a  mystifying  conversion  of  very 
simple  linear  ratios  into  a guise  of  root-rectangles.”  There  is  no  such  thing  as 
magic.  The  word  is  a euphemism  for  such  terms  as  quackery,  charlatanism,  which 
imply  deliberate  or  unconscious  deception  on  the  part  of  the  manipulator.  The 
only  opportunity  for  deception  in  the  present  case  is  in  the  recording  of  the 
dimensions  of  the  vases.  And,  so  far  as  I know,  no  one  has  yet  charged  Mr. 
Hambidge  and  his  collaborators  with  deliberate  falsification  of  the  evidence. 

5.  Before  the  publication  of  Mr.  Hambidge’s  book  the  question  whether  Attic 
potters  constructed  their  vases  according  to  a predetermined  design  had  hardly 
been  raised.  In  his  catalogue  of  the  Attic  pottery  in  the  Louvre,  published  in 
1906,  M.  Pottier  had  indeed  remarked:  “Les  proportions  des  vases,  les  rapports 
de  mesures  entre  les  differentes  parties  de  la  poterie  paraissent  avoir  ete  chez  les 
Grecs  l’objet  de  recherches  minutieuses  et  delicates  . . . Je  crois  qu’un  examen 
attentif  du  sujet  menerait  a des  observations  interessantes  sur  ce  qu’on  pourrait 
appeler  la  ‘geometrie’  de  la  ceramique  grecque”  ( Vases  antiques,  III,  p.  658). 
Apparently  Mr.  Hambidge  was  the  first  to  undertake  such  a careful  examination 
of  the  subject.  It  could  only  be  done  on  the  basis  of  accurate  measurements  of  all 
the  elements  of  a large  number  of  vases;  and  the  labor  of  procuring  these  measure- 

[ 32  ] 


INTRODUCTION 


ments  did  not  seem  worth  while  so  long  as  no  method  of  coordinating  the  propor- 
tions had  been  discovered.  That  the  geometric  analysis  of  the  vases  in  terms  of 
the  root-rectangles  furnishes  such  a method  can  hardly  be  disputed.  If  the  pro- 
portions thus  revealed  cannot  be  satisfactorily  explained  either  as  coincidences,  or 
as  “a  mystifying  conversion  of  very  simple  linear  ratios  into  a guise  of  root- 
rectangles,”  the  only  alternative  is  to  suppose  that  the  potters  worked  from  a 


Diagram  XLIV 


drawing.  This  is  perhaps  less  improbable  than  it  appears  to  Mr.  Carpenter  when 
he  asks:  “Must  we  hold  that  the  sixth  and  fifth  century  potters  — often  slave- 
born  humble  artisans  — knew  all  this  geometry,  and,  since  the  constructions  can- 
not be  done  mentally  nor  yet  on  the  potter’s  table,  used  up  precious  parchment  to 
draw  these  rectangles  and  diagonals?  Or  are  we  to  assume  that  they  derived  their 
measurements,  correct  to  a fairly  small  fraction  of  an  inch,  from  contemplating 
figures  drawn  Archimedean-wise  with  a pointed  stick  on  the  ground,  or  with  char- 
coal on  a slab  of  wood.”  Slaves  in  Athens  were  not  necessarily  always  inferior 

[ 33  ] 


GEOMETRY  OF  GREEK  VASES 


intellectually  to  Athenian  citizens.  It  is  conceivable  also  that  the  graphic  methods 
of  dynamic  symmetry  might  appeal  more  to  Nikosthenes,  Euphronios,  and 
Brygos  than  the  feats  of  mental  arithmetic  which  appear  simpler  to  Mr.  Carpen- 
ter. Nor  need  we  assume  so  lavish  a consumption  of  precious  parchment.  A slab 
of  Pentelic  marble  would  serve  admirably  both  as  drawing  board  and  paper  for  a 
thousand  designs.  I recently  saw  such  a slab  on  the  Acropolis  which  was  evidently 
being  put  to  a similar  use  in  connection  with  plans  for  repairs  of  the  ancient  build- 
ings. That  Attic  potters  as  well  as  architects  could  use  a T square  and  right 
angle  does  not  seem  fantastic.  After  a full-size  drawing  was  made,  it  would  be 
simple  to  control  all  the  diameters  of  a vase  by  means  of  calipers,  and  all  the 
heights  by  means  of  a stick  thrust  horizontally  into  a lump  of  clay  set  up  beside 
the  potter’s  wheel.  Moreover,  it  is  well  to  remind  ourselves  that  it  is  shnpler  for 
an  artist  to  produce  a design  in  terms  of  the  root-rectangles  than  for  an  investiga- 
tor working  two  thousand  years  later  to  recover  the  design  and  the  exact  method 
by  which  it  was  produced.  Finally,  we  may  ask  whether  it  would  be  practicable 
to  make  a vase  like  the  krater,  no.  73,  which  is  two  and  a half  feet  high,  and  com- 
posed of  at  least  seven  different  pieces,  without  the  help  of  an  accurate  drawing. 
Certainly  no  potter  of  the  present  day  would  attempt  such  a feat. 


J 


[34] 


ANALYSES  OF  VASES 


AMPHORA 

BLACK-FIGURED  PERIOD 

The  collection  contains  about  forty  amphorae  with  black-figured  decoration 
which  are  sufficiently  well-preserved  to  be  studied  from  the  point  of  view  of  pro- 
portions. Twenty-six  of  these  have  been  selected  for  publication.  Nos.  1-21 
are  “ neck  amphorae,”  of  several  types,  so-called  because  the  division  between 
neck  and  shoulder  is  clearly  marked.  Nos.  22-26  are  “ panel  amphorae,”  with- 
out division  between  neck  and  shoulder,  and  with  the  painted  decoration  placed 
in  panels,  one  on  each  side  of  the  vase.  No  satisfactory  analysis  of  the  neck 
amphorae,  nos.  2,  3,  6,  14,  20,  21  has  been  found,  nor  of  any  of  the  panel  am- 
phorae, except,  perhaps,  no.  22.  Nos.  14,  21,  23,  24  have  the  ratio  1.500,  i.  e., 
their  height  equals  l|-  times  their  diameter.  They  belong  apparently  to  the 
“static”  type  of  symmetry.  In  several  cases,  especially  nos.  16  and  19,  it  has 
been  necessary  to  depart  considerably  from  the  actual  dimensions  in  order  to 
obtain  an  intelligible  analysis.  Slightly  more  than  fifty  per  cent  of  the  amphorae 
studied  are  thus  seen  to  have  proportions  which  can  be  expressed  accurately  in 
terms  of  “dynamic  symmetry.”  The  chief  proportions,  so  far  as  they  have  been 
ascertained,  are  given  in  the  table  of  ratios  on  the  following  page. 


[ 35  ] 


GEOMETRY  OF  GREEK  VASES 


No. 

Type  of  Amphora 

Height 

Height 
of  neck 
and  lip 

Height  to 
shoulder 

Width 

Diameter 
of  lip 

Diameter 

of 

shoulder 

Diameter 
of  bottom 
of  body 

Diameter 
of  foot 

1 

Neck  amphora 

1.2034 

.2927 

.9107 

1.000 

.691 

.559 

.333 

.559 

2 

1.2071 

.2735 

.9336 

1.000 

.5858? 

.4531 

.3137 

.500 

3 

1.288 

1.000 

4 

1.382 

.3618 

1.0202 

1.000 

.679 

.5528 

.333 

.5528 

5 

1.382 

.309 

1.073 

1.000 

.5801 

.382 

.236 

.4811 

6 

1.4045 

1.000 

.618 

7 

1.4472 

.382 

1.0652 

1.000 

.6584 

.500 

.382 

.6584 

8 

1.4472 

.382 

1.0652 

1.000 

.6584 

.500 

.309 

.528 

9 

1.472 

.354 

1.118 

1.000 

.708 

.472 

.333 

.5669 

10 

1.472 

.382 

1.090 

1.000 

.6584 

.4472 

.333 

.5394 

11 

1.472 

.4045 

1.0675 

1.000 

.708 

.4472 

.309 

.500+ 

12 

1.472 

.3455 

1.265 

1.000 

.618 

.4472 

.250 

.500 

13 

1.472 

.368 

1.104 

1.000 

.618 

.500 

.309 

.500 

14 

1.500 

.375 

1.125 

1.000 

15 

1.528 

.3618 

1.1662 

1.000 

.618 

.4045 

.236? 

.528? 

16 

1.618 

.416 

1.000 

1.000 

.6584 

.472 

.250? 

.500 

17 

1.618 

.500 

1.118 

1.000 

.7764 

.528 

.236 

.5669 

18 

1.618 

.382 

1.236 

1.000 

.618 

.4472 

.309 

.5528 

19 

1.618 

.382 

1.236 

1.000 

.6584 

.500 

.333 

.500 

20 

Panathenaic 

1.4142 

.2357 

1.1785 

1.000 

.333 

.1716 

21 

1.500 

.250 

1.250 

1.000 

22 

Panel  amphora 

1.4142 

1.000 

.6095 

.333 

.5286 

23 

1.500 

1.000 

24 

1.500 

1.000 

25 

1.528 

1.000 

26 

1.5773 

1.000 

.6512 

.333 

.5118 

1 Neck  Amphora.  Inv.  98.923.  ( A and  B.)  Combat  between  two  warriors. 
Height,  0.293  m.  Width,  handle  to  handle,  0.245  m.  Diameter  of  bowl, 
0.240  m.  Diameter  of  lip,  0.168  m.  Diameter  of  foot,  0.136  m. 

The  ratio  is  1.2034,  to  obtain  which  the  width  must  be  assumed  to  be  0.24345  m. 
The  resulting  rectangle  can  be  expressed  in  various  ways,  the  simplest  being 
.309+. 4472+. 4472.  The  geometrical  analysis  is  not  as  simple  as  in  most  cases. 
Study  of  the  details  shows  that  most  of  them  are  derived  from  the  V 5 rectangle 
rather  than  from  the  whirling  square  rectangle.  In  the  drawing  .309  rectangles 
have  been  applied  at  the  top  and  bottom  of  the  containing  area. 

The  ratios  are : 

Height 1.2034 

Height,  omitting  lip 1.118 

Height , omitting  foot 1.118 


= .2927  + .618  + .2927 

= y'1 * * * 5 
2 

= V5 
2 


[ 36  ] 


AMPHORA 


Height  to  shoulder 

Height  of  lip  and  of  foot . . . 
Width 

Diameter  of  lip 

Smallest  diameter  of  neck . . 

Diameter  of  top  of  neck.  . . . 

Diameter  of  shoulder 

Diameter  of  bottom  of  body 
Diameter  of  foot 


.9107 

= .618  + .2927 

.0854 

1.000 

= 1 

.691 

1 5 

1 + V5 

.500 

1 

2 

.559 

V5 

4 

.559 

_ V5 
4 

.333 

1 

— 3 

.559 

V5 

4 

GEOMETRY  OF  GREEK  VASES 


2 Neck  Amphora.  Inv.  89.273.  On  shoulder  ( A and  B)  a cock  fight.  On 
body  ( A ) departure;  ( B ) return  of  a warrior.  Below,  zone  of  animals. 

Height,  0.4395  m.  Diameter,  0.365  m. 

These  dimensions  give  approximately  the  rectangle  1.2071,  composed  of  half  a 
square  (.500)  plus  a \/2  rectangle  (.7071).  The  proportions  of  details,  however, 
are  not  all  accurately  expressible  in  simple  terms  of  this  rectangle.  In  the  drawing 
the  two  squares  have  been  placed  above  the  \/2  rectangle.  The  height  and 
diameter  of  the  shoulder  are  fixed  by  the  intersection  of  the  diagonal  of  one  of 
the  squares  with  the  diagonal  of  the  containing  rectangle.  The  height  of  the 
lip  and  neck  is  .27346,  the  height  to  the  shoulder  .9336,  the  diameter  of  the 
shoulder  .4531.  The  nearest  simple  ratio  for  the  diameter  of  the  lip  is  .5858,  or 
1.000 --.4142.  But  this  is  wrong  by  5 millimetres  as  is  shown  by  the  geometrical 
construction.  The  diameter  of  the  foot  is  close  to  half  the  diameter  of  the  vase. 
The  diameter  of  the  bottom  of  the  body  is  obtained  from  the  intersection  of  the 
diagonal  of  the  whole  with  the  top  of  a .4142  rectangle  ( = 2.4142,  or  a square  plus 
a v/2  rectangle)  applied  at  the  bottom.  The  ratio  of  the  diameter  of  the  bottom  of 
the  body  is  .3137.  The  analysis  is  unsatisfactory. 


C 38  ] 


AMPHORA 


3 


3 Double-bodied  Neck  Amphora.  Inv.  00.331.  Annual  Report,  1900,  p.  33, 
no.  2.  On  shoulder,  two  battle-scenes.  On  body  (A)  Theseus  and  the  Minotaur; 
(. B ) Departure  of  a warrior.  Below,  boys  on  horseback  and  warriors  on  foot. 

Height,  0.358  m.  Diameter,  0.278  m. 

The  ratio  is  1.288,  or  .7764.  The  latter  is  composed  of  .500  plus  .2764,  which  is 
the  reciprocal  of  3.618,  or  two  squares  plus  a whirling  square  rectangle.  If  this 
3.618  shape  is  applied  at  each  side,  the  space  in  the  centre  is  made  up  of  two  y/5 
rectangles.  This  central  area  determines  the  diameter  of  the  bottom  of  the  body, 
and  accounts  for  most  of  the  details  of  neck  and  foot.  The  diameter  of  the  lip  is 
somewhat  more  than  double  that  of  the  bottom  of  the  body.  It  is  determined 
exactly  by  applying  vertical  whirling  square  rectangles  at  both  sides,  the  lip  being 
contained  in  the  overlapping  portion.  The  rectangle  1.288  occurs  rarely;  and  the 
geometrical  analysis  is  not  simple  enough  to  be  convincing. 


C 39  ] 


GEOMETRY  OF  GREEK  VASES 


4 Neck  Amphora.  Inv.  01.8026.  Arch.  Zeitung,  1884,  pi.  III.  (A)  Two 
warriors.  (B)  Athena  in  conversation  with  a draped  bearded  figure.  Signed  by 
Amasis  as  potter. 

Height,  0.258  m.  Diameter,  0.1865  m. 

These  dimensions  give  very  accurately  the  ratio  1.382.  This  may  be  divided 
into  a square  and  a .382  rectangle,  or  into  two  .691  rectangles,  each  composed  of  a 
square  and  a y/h  rectangle.  If  a square  be  applied  at  the  bottom  of  the  contain- 
ing area,  its  top  coincides  with  a painted  band  below  the  shoulder.  The  inter- 
section of  the  diagonal  of  the  whole  shape  with  the  diagonal  of  a square  applied 
at  the  end  of  the  .382  rectangle  determines  the  diameter  of  the  lip.  The  diameter 
of  the  bottom  of  the  body  is  one-third  of  the  total  diameter.  Most  of  the  other 
proportions  are  obtained  from  subdivisions  of  the  y/5  rectangle  applied  to  the 
two  .691  rectangles  which  make  up  the  whole  area. 


Height 1.382 

Height  to  shoulder 1.0202 

Height  of  neck  and  lip 3618 

Diameter.  . 1.000 

Diameter  of  lip 679 


Diameter  of  shoulder 5528 

Diameter  of  bottom  of  body . 333 

Diameter  of  top  of  foot 500 

Greatest  diameter  of  foot 5528 


C 40  ] 


AMPHORA 


[ 41  ] 


GEOMETRY  OF  GREEK  VASES 


5 Neck  Amphora.  Inv.  99.  517.  On  the  neck  and  the  upper  part  of  the  body, 
groups  of  figures  painted  in  the  “affected  style.”  The  lower  part  of  the  body  is 
black. 

Height,  0.4715  m.  Diameter,  0.3435  m. 

To  obtain  the  ratio  1.382  the  height  is  assumed  to  be  0.473  m.  and  the  diameter 
0.34225  m. 

The  proportions  of  lip  and  foot  are  obtained  from  the  diagonal  of  the  contain- 
ing shape.  The  diameter  of  the  lip  is  given  by  the  intersection  of  this  diagonal 
with  the  diagonal  of  a square  applied  at  the  upper  right  hand  corner  of  the  large 
rectangle.  Ratio  .580.  The  diameter  of  the  foot  is  fixed  by  the  intersection  of  the 
diagonal  of  the  whole  shape  with  the  diagonal  of  the  applied  square  of  a whirling 
square  rectangle  inscribed  on  the  base  (ratio:  .48113).  The  geometrical  con- 
structions for  determining  the  height  of  the  neck  and  lip  (.309),  the  diameter  of 
the  shoulder  (.382),  and  the  diameter  of  the  bottom  of  the  body  (.236)  are  simple. 

[ 42  ] 


AMPHORA 


6 Neck  Amphora.  Inv.  01.8035.  (A)  Shoemaker’s  shop.  ( B ) Blacksmith’s 

shop. 

Height,  0.365  m.  Diameter,  0.256  m. 

The  ratio  is  apparently  1.4045;  but  no  satisfactory  analysis  has  been  found. 
The  diameter  of  the  lip  has  the  ratio  .618.  An  attempt  to  fit  the  amphora  into  a 
y/2  rectangle  (1.4142)  also  proved  unsuccessful. 


[ 43  ] 


GEOMETRY  OF  GREEK  VASES 


7 Neck  Amphora.  Inv.  01.8027.  (A)  Hauser,  Jahreshefte,  1907,  pp.  1 ff., 
pi.  I-IV.  Herakles  and  Apollo  fighting  for  the  tripod.  ( B ) Thetis  bringing  the 
armor  to  Achilles.  Signed  by  Amasis  as  potter. 

Height,  as  restored,  0.306  m.  (given  by  Hauser  as  0.31  m.).  Diameter,  0.213- 
0.215  m.  The  exact  height  is  uncertain,  since  a small  portion  of  the  vase  near  the 
bottom  of  the  body  is  apparently  lacking.  The  restoration  cannot,  however,  have 
changed  the  height  more  than  1 or  2 mm.  The  ratio  is  assumed  to  be  1.4472. 
If  the  diameter  be  taken  as  0.213  m.,  the  height  to  give  this  ratio  should  be 
0.30825  m.  With  these  main  dimensions  the  ratios  of  details  work  out  very  ac- 
curately. The  height  of  the  neck  and  lip  is  .382,  the  height  to  the  shoulder, 
1.0652  (.618+.4472).  The  diameter  of  the  lip  equals  that  of  the  foot;  both  have 
the  ratio  .6584.  The  diameter  of  the  shoulder  is  .500.  The  diameter  of  the  bottom 
of  the  body  is  .382.  Various  other  details  are  simply  obtained  by  geometrical 
construction,  as  appears  in  the  drawing. 

[ 44  ] 


AMPHORA 


8 Neck  Amphora.  Inv.  76.41.  (A)  Herakles  and  the  Nemean  lion.  (B) 
Dionysos  between  two  maenads. 

Height,  0.3735  m.  Diameter,  0.258  m. 

The  ratio  is  1.4472.  The  containing  area  is  made  up  of  a square  plus  a \/5 
rectangle.  The  main  proportions  of  the  upper  part  of  the  amphora  are  exactly 
like  those  of  the  amphora  decorated  by  Amasis,  no.  7,  which  is  contained  in  the 
same  rectangle.  The  height  of  neck  and  lip  is  .382,  the  height  to  the  shoulder 
1.0652  (.618 +.4472).  The  diameter  of  the  lip  is  .6584,  that  of  the  shoulder  .500. 
The  proportions  of  the  foot  are  different,  but  equally  simple;  the  diameter  of  the 
bottom  of  the  body  is  .309,  that  of  the  foot,  .528.  These  last  two  ratios  are  more 
nearly  normal  than  the  corresponding  ratios  in  the  preceding  amphora,  which  is 
perhaps  unique  in  having  its  foot  as  large  as  its  lip. 


[ 45  ] 


GEOMETRY  OF  GREEK  VASES 

9 Neck  Amphora.  Inv.  76.40.  (A)  Departure  of  a warrior  in  a quadriga. 
(B)  Dionysos,  Ariadne,  seilens  and  maenads.  The  body  is  red. 

Height,  0.298  m.  Diameter,  0.20175  m. 

To  obtain  the  ratio  1.472  the  diameter  is  assumed  to  be  0.2025  m.  The  pro- 
portions are  all  expressible  in  simple  ratios.  The  containing  shape  may  be  divided 

into  1.000  + 236  + .236,  the 
square  being  placed  at  the  bot- 
tom. The  vase  to  the  shoulder 
is  enclosed  in  the  rectangle 
1.118,  the  line  of  the  shoulder 
coming  at  half  the  height  of 
the  lower  of  the  two  .236  rec- 
tangles. It  is  also  noteworthy 
that  the  height  of  the  foot  is 
.118  and  that  consequently  the 
body  of  the  vase  is  enclosed 
in  a square.  The  proportions 
of  the  neck  and  lip  are  clearly 
expressed  in  terms  of  the  .236 
rectangle  applied  at  the  top  of 
the  containing  area.  This  is 
divided  into  three  squares 
flanked  by  whirling  square 
rectangles.  The  lip  is  con- 
tained in  the  three  squares 
(.236  X 3 = .708) . The  diameter 
of  the  shoulder  is  .708  — .236, 
or  .472.  The  height  of  the  neck 
and  lip  is  .236 +.118,  or  .354. 
This  is  half  of  .708.  The  propor- 
tions of  the  foot  are  obtained 
from  divisions  of  a whirling 
square  rectangle  applied  at 
the  bottom  of  the  enclosing  area.  The  diameter  of  the  foot  is  fixed  by  the  in- 
tersection of  the  diagonal  of  half  the  whirling  square  rectangle  with  the  diagonal 
of  its  reciprocal  (.5669).  The  diameter  of  the  bottom  of  the  body  is  one-third 
of  the  diameter  of  the  amphora,  expressed  geometrically  in  the  drawing  by  the 
intersection  of  the  diagonal  of  the  whirling  square  rectangle  with  the  diagonal 
of  its  half. 


[ 46  ] 


AMPHORA 


10  Neck  Amphora.  Inv.  80.621.  (A)  Combat  with  a sea  monster.  (J3) 
Dionysiac  scene. 

Height,  0.3768  m.  Diameter,  0.256  m. 

The  ratio  is  1.472.  In  the  drawing  a whirling  square  rectangle  has  been  ap- 
plied at  the  top  of  the  containing  shape.  It  will  be  readily  seen  that  the  neck  and 
lip  are  contained  in  a .382  rectangle.  The  height  of  the  amphora  to  the  shoulder  is, 
consequently,  1.472  — .382,  or  1.090.  The  diameter  of  the  lip  has  the  ratio  .6584, 
expressed  geometrically  by  the  intersection  of  the  diagonal  of  half  the  whirling 
square  rectangle  with  the  diagonal  of  a square  applied  at  the  end  of  the  .382 
rectangle.  The  diameters  of  the  shoulder  and  the  top  of  the  neck  have  the 
familiar  ratio  .4472.  The  diameter  of  the  bottom  of  the  body  is  one-third  that  of 
the  amphora.  The  diameter  of  the  foot  has  the  ratio  .5394,  which  is  the  reciprocal 
of  1.854,  or  three  whirling  square  rectangles.  These  two  proportions  are  expressed 
geometrically  in  the  drawing  by  applying  a 1.854  rectangle  at  the  bottom  of  the 
containing  area.  The  bottom  of  the  body  is  contained  in  the  central  whirling 
square  rectangle.  The  diameter  of  the  foot  is  fixed  by  the  intersection  of  the 
diagonal  of  half  a square  with  the  top  of  the  1.854  rectangle. 


[47] 


GEOMETRY  OF  GREEK  VASES 


11  Neck  Amphora.  Inv.  89.258.  On  the  shoulder,  between  two  large  eyes 
( A ) Athena  driving  a quadriga;  (B)  Bearded  man  driving  a quadriga.  The 
body  is  black. 

Height,  0.4034  m.  Diameter,  0.2735  m. 

The  ratio  is  1.472.  The  proportions  of  the  upper  part  can  be  obtained  simply 
from  the  whirling  square  rectangle  applied  at  the  top  of  the  containing  area. 
The  ratio  of  the  diameter  of  the  lip  (.708)  is  shown  by  subdivisions  of  the  recip- 
rocal of  this  whirling  square  rectangle.  The  diameter  of  the  shoulder  is  con- 
tained in  the  central  square  of  a y/h  rectangle  applied  to  the  whirling  square 
rectangle  (.4472).  The  height  of  the  neck  and  lip  is  contained  in  four  vertical 
whirling  square  rectangles  (.4045  = 1.618-^-4).  The  height  to  the  shoulder  is 
consequently  1.0675.  The  diameter  of  the  foot  is  somewhat  more  than  .500.  The 
diameter  of  the  bottom  of  the  body  is  .309. 


[ 48  ] 


AMPHORA 


12  Neck  Amphora.  Inv.  99.516.  On  the  neck  and  the  upper  part  of  the  body, 
groups  of  figures  painted  in  the  “ affected  style.”  The  lower  part  of  the  body  is 
black.  Height,  0.298  m.  Diameter,  0.20175  m. 

The  ratio  is  1.472.  The  proportions  of  the  upper  part  of  the  amphora  are 
obtained  from  the  whirling  square  rectangle  applied  at  the  top  of  the  containing 
area.  The  intersection  of  the  diagonals  of  the  reciprocal  of  this  whirling  square 
rectangle  determines  the  diameter  of  the  lip  (.618).  The  diameter  of  the  shoulder 
is  contained  exactly  in  the  central  square  of  a \/5  rectangle  applied  to  the  whirl- 
ing square  rectangle  (.4472).  If  a square  be  applied  in  the  centre  of  the  whirling 
square  rectangle,  the  intersection  of  its  diagonal  with  the  diagonal  of  the  central 
square  of  the  \/5  rectangle  fixes  the  height  of  the  shoulder  (1.265).  The  height  of 
the  neck  and  lip  is  .3455,  or  .691  -j-  2.  The  proportions  of  some  other  details  are 
simply  obtained,  e.  g.,  the  height  of  the  lip  (.0854).  The  area  above  the  painted 
bands  round  the  middle  of  the  body  is  made  up  of  two  whirling  square  rectangles 
(.809).  The  area  below  the  painted  bands  is  a whirling  square  rectangle  (.618). 
The  diameters  of  the  bottom  of  the  body  and  the  foot  have  the  simple  ratios  .250 
and  .500. 


[ 49  ] 


GEOMETRY  OF  GREEK  VASES 


13  Neck  Amphora.  Inv.  105.09.  (A)  Athena  mounting  a quadriga,  Apollo, 
Hermes.  (B)  Combat  between  two  warriors;  a bearded  man  comes  between  them. 

Height,  0.3885  m.  Diameter,  0.265  m. 

If  the  diameter  is  assumed  to  be  0.264  m.,  the  ratio  is  1.472.  To  obtain  the 
proportions  of  the  upper  part  of  the  amphora,  a whirling  square  rectangle  is 
applied  at  the  top  of  the  containing  area.  The  intersection  of  the  diagonals  of  the 
reciprocal  of  this  whirling  square  rectangle  determines  the  diameter  of  the  lip 
(.618).  The  diameter  of  the  top  of  the  neck  has  the  ratio  .4472.  The  smallest 
diameter  of  the  neck  is  obtained  by  the  intersection  of  the  diagonals  of  the 
applied  square  of  the  whirling  square  rectangle  (.382).  The  diameter  of  the 
shoulder  and  its  height  are  fixed  by  the  intersection  of  the  diagonal  of  the  whole 
shape  with  a diagonal  of  the  same  applied  square.  The  diameter  of  the  shoulder 
has  the  ratio  .500  (actually  slightly  less).  This  is  also  the  ratio  of  the  diameter  of 


[ 50  ] 


AMPHORA 


the  foot.  The  height  to  the  shoulder  has  the  ratio  1.104,  which  is  three-quarters 
of  1.472.  The  diameter  of  the  bottom  of  the  body  has  the  ratio  .309;  and  details 
of  the  foot  can  be  obtained  from  simple  subdivisions  of  a .309  rectangle  applied  at 
the  bottom  of  the  enclosing  area. 

14  Neck  Amphora.  Inv.  01.8052.  (A)  Dionysos  reclining,  with  a female 
figure  seated  before  him,  and  seilens  to  right  and  left.  (5)  Seilens  and  a maenad 
in  a vineyard. 

Height,  0.4265  m.  Diameter,  0.284  m. 

The  enclosing  rectangle  has  the  ratio  1.500.  None  of  the  dimensions  of  details 
are  expressible  in  simple  terms  of  the  root  rectangles  or  the  rectangle  of  the  whirl- 
ing squares.  The  proportions  are  probably  of  the  “ static  ” class;  but,  with  the 
exception  of  the  height  of  the  shoulder,  which  is  three-quarters  of  the  total 
height,  no  simple  ratios  have  been  found. 


[ 51  ] 


GEOMETRY  OF  GREEK  VASES 


15  Neck  Amphora.  Inv.  13.76.  Hambidge,  p.  91.  On  the  neck  ( A and  B ) 
four  figures  in  the  “ affected  style.”  The  body  is  black. 

Height,  0.4105  m.  Diame- 
ter, 0.27  m. 

If  the  diameter  be  assumed 
to  be  0.269m.,  the  amphora  fits 
the  rectangle  1.528.  The  pro- 
portions of  details  are  most 
simply  expressed  in  terms  of 
whirling  square  rectangles. 
The  height  of  the  neck  and 
lip  has  the  ratio  .3618.  This  is 
obtained  from  a \/5  rectangle 
applied  at  the  top  of  the  con- 
taining area.  A whirling  square 
rectangle  is  applied  to  this  y/5 
rectangle  (at  the  right  in  the 
drawing),  and  its  reciprocal  is 
bisected,  giving  the  level  of  the 
shoulder.  The  diameter  of 
the  shoulder  is  determined  by 
the  intersection  of  this  line 
with  the  diagonal  of  half  the 
large  whirling  square  rec- 
tangle. The  ratio  is  .4045  or 
1.618-7-4.  The  diameter  of  the 
lip  is  .618;  the  diameter  of 
the  top  of  the  neck  is  .4472; 
the  smallest  diameter  of  the 
neck  (not  shown  geometrically 
in  the  drawing)  is  .382.  The 
whirling  square  rectangle  applied  at  the  bottom  of  the  area  accounts  for  the 
diameter  of  the  bottom  of  the  body  (.236)  and  of  the  foot  (.528  — ).  It  is  pos- 
sible, however,  that  these  diameters  were  intended  to  be  .250  and  .500,  as  in  the 
amphora,  no.  12. 


[ 52  ] 


AMPHORA 


16  Neck  Amphora.  Inv.  01.17.  Amphora  with  high,  volute  handles,  and 
black-figured  decoration  on  a white  ground.  Dionysos,  riding  on  a mule,  ac- 
companied by  seilens  and 
maenads. 

Height  of  handles,  0.29  m. 

Height  of  lip,  0.253  m.  Height 
to  shoulder,  0.18  m.  Great- 
est width  (handles),  0.1835  m. 

Diameter  of  bowl,  0.153  m. 

Diameter  of  lip,  0.129  m.  Di- 
ameter of  foot,  0.09  m. 

This  vase  does  not  fit  ac- 
curately any  simple  rectangle. 

It  is  noteworthy,  however, 
that  the  diameter  of  the  foot 
equals  half  the  height  to  the 
shoulder.  This  dimension, 

0.18  m.,  multiplied  by  1.618, 
gives  0.29124  m.  If  the  great- 
est width  is  reduced  to  0.18m., 
the  amphora  can  be  placed  in 
a whirling  square  rectangle, 
and  the  amphora  up  to  the 
shoulder  in  a square  applied 
at  the  bottom.  Most  of  the 
details  are  expressible  in  simple 
terms  of  the  whirling  square 
rectangle,  as  follows : 


Height  of  handles 

Height  of  lip 

Height  to  shoulder 

Width 

Diameter  of  bowl 

Diameter  of  lip 

Diameter  of  shoulder 

Diameter  of  bottom  of  body 
Diameter  of  foot 


1.618 
1.416  = 
1.000 
1.000  + 
.854 
.6584 


.236  X 6 


.472  = .236  X 2 

.250 

.500 


[ 53  ] 


GEOMETRY  OF  GREEK  VASES 


17  Neck  Amphora.  Inv. 

95.829.  Dionysos,  seated.  Poor 
drawing.  Height,  0.1695  m. 
Diameter,  0.104  m. 

The  ratio  is  close  to  1.618. 
All  the  details  are  in  simple 
terms  of  the  whirling  square 


rectangle,  as  follows: 

Height 1.618 

Height  to  shoulder 1.118 

Height  of  neck  and  lip  . . . .500 

Height  of  neck 382 

Height  of  lip 118 

Diameter 1.000 

Diameter  of  lip 7764 

Diameter  of  shoulder 528 

Diameter  of  bottom  of 

body 236 

Diameter  of  foot 5669 


18  Neck  Amphora.  Inv.  01.8059.  Hambidge,  p.  129.  ( A and  B)  Gigan- 
tomachy. 

Height,  0.3425  m.  Diameter,  0.213  m. 

The  ratio  is  close  to  1.618,  and  the  proportions  are  clearly  expressed  in  terms  of 
the  whirling  square  rectangle,  as  is  shown  by  the  geometrical  analysis  and  by  the 
following  table  of  ratios: 


Height 

Height  to  shoulder 

Height  of  neck  and  lip 

Diameter 

Diameter  of  lip 

Diameter  of  shoulder 

Diameter  of  bottom  of  body . 
Diameter  of  foot 


1.618 

1.236  = .618  + .618 
.382  = 1.000  - .618 


1.000 

.618 


.4472  = 


V5 


.309  = .618  ^ 2 


.5528  = 1.000  - .4472 


[ 54  ] 


AMPHORA 


[55  ] 


GEOMETRY  OF  GREEK  VASES 


19 


19  Neck  Amphora.  Inv.  98.916.  Annual  Report,  1898,  p.  59,  no.  25.  Decora- 
tion in  three  zones:  1.  (A)  Herakles  and  Amazons.  ( B ) Men  dancing,  between 
two  swans.  2.  Sirens,  sphinx,  panthers,  goat.  3.  Panthers,  ram,  swan.  This  is  the 
well-known  amphora  of  early  Attic  (“Tyrrhenian”)  style,  found  at  Vulci,  and 
formerly  in  the  Tyskiewicz  collection.  Gsell,  Fouilles  de  Vulci,  p.  118,  pi.  V,  VI. 

Height,  0.394  m.  Diameter,  0.2405  m.  Diameter  of  shoulder,  0.122  m.  Diam- 
eter of  foot,  0.121  m. 

If  the  diameter  is  taken  as  0.242  m.,  or  twice  the  diameter  of  the  foot,  and  the 
height  as  0.3916  m.,  the  amphora  is  contained  in  a whirling  square  rectangle 
(ratio:  1.618).  The  height  of  the  shoulder  has  very  nearly  the  ratio  1.236.  The 
diameter  of  the  bottom  of  the  body  is  one-third  of  the  largest  diameter  (.333) ; 
that  of  the  shoulder  and  of  the  foot  one-half  of  the  largest  diameter  (.500).  The 
diameter  of  the  lip  has  the  ratio  .6584,  obtained  by  applying  a .1708  rectangle  at 
the  top,  and  subtracting  a square  from  either  end  of  it.  Cf.  diagram  XIII,  p.  10. 

[56] 


AMPHORA 


20  Panathenaic  Prize  Amphora.  Inv.  01.8127.  (A)  Athena  with  spear  and 
shield,  (j B)  Boxing  match.  Sixth  century,  b.c. 

Height,  0.598  m.  Diameter,  0.421  m. 

These  dimensions  give  the  ratio  1.4142,  or  s/2.  The  diameter  of  the  shoulder 
is  one-third  of  the  total  diameter.  The  height  to  the  shoulder  is  five-sixths  of  the 
total  height.  The  diameter  of  the  lip  is  obtained  by  a fairly  simple  geometrical 
division  of  the  area  above  the  shoulder,  which  is  composed  of  three  \/2  rec- 
tangles placed  horizontally  side  by  side.  The  diameter  of  the  bottom  of  the  body 
is  obtained  by  applying  squares  at  either  end  of  a .4142  rectangle  ( = \/2  — 1) 
applied  to  the  bottom  of  the  containing  area.  The  diameter  of  the  foot  is  not  ex- 
pressible in  simple  terms  of  the  -y/2  rectangle. 

21  Panathenaic  Prize  Amphora.  Inv.  99.520.  ( A ) Athena  with  spear  and 
shield.  (B)  Foot-race.  Sixth  century,  b.c. 

Height,  0.61  m.  Diameter,  0.404  m. 

If  the  diameter  is  taken  as  0.405  m.  and  the  height  as  0.6075  m.  the  ratio  is 
1.500,  or  a square  and  a half.  The  height  of  the  shoulder  has  nearly  the  ratio 
1.250,  or  a square  and  a quarter.  No  other  simple  ratios  have  been  found. 


[ 57  ] 


GEOMETRY  OF  GREEK  VASES 


22  Panel  Amphora. 

Inv.  00.330.  (A)  Birth 
of  Athena.  (B)  Quadriga, 
in  front  view. 

Height,  0.393  m.  Width 
(handles),  0.275  m.  Di- 
ameter of  body,  0.268  m. 

The  analysis  of  this 
vase  is  doubtful.  In  the 
drawing  it  has  been  en- 
closed in  a y/2  rectangle, 
ratio  1.4142.  To  obtain 
this  ratio  3 mm.  must 
be  added  to  the  width, 
assuming  the  average 
height,  0.393  m.,  to  be 
correct.  The  error  can 
also  be  distributed,  mak- 
ing the  height  0.391  m., 
the  width  0.2764  m.  No 
other  simple  ratio  has 
been  found  which  fits 
the  proportions  more  ac- 
curately; and  the  details 
work  out  satisfactorily 
in  terms  of  the  y/2  rec- 
tangle. The  proportions 
of  lip  and  foot  are  shown 
by  subdivisions  of  one  of 
three  \/2  rectangles  applied  at  the  top  and  bottom  of  the  over-all  area.  The 
chief  ratios  are: 


22 


Height 

Width 

Diameter  of  lip 

Diameter  of  bottom  of  body 
Diameter  of  foot 


1.4142  = y/2 


1.000  = 1 

,6095  _ ;v2pi 

O 

.333  = i 


.5286  = 1.000  - .4714  = 


3 - V2 


[58] 


AMPHORA 


23  Panel  Amphora.  Inv.  98.918.  (A)  Girl  seated  in  a swing.  ( B ) Winged 
female  figure  running  between  bearded  male  figures. 

Height,  0.402  m.  Width  (handles),  0.268  m.  Diameter  of  body,  0.262  m. 

The  containing  rectangle  has  the  ratio  1.500.  Like  the  neck  amphora,  no.  14, 
this  is  apparently  an  example  of  “ static  symmetry.”  No  simple  ratios  have  been 
determined. 

24  Panel  Amphora.  Inv.  01.8053.  In  both  panels,  Dionysos,  Hermes,  seilens 
and  other  figures,  painted  in  the  “ affected  style.” 

Height,  0.453  m.  Width  (handles),  0.3005  m.  Diameter  of  body,  0.277  m. 

The  ratio  of  the  containing  rectangle  is  close  to  1.500.  No  satisfactory  analysis 
has  been  found. 


[ 59  ] 


GEOMETRY  OF  GREEK  VASES 


25  Panel  Amphora.  Inv.  99.538.  ( A ) Herakles  and  the  Cretan  bull,  in  the 
black-figured  style.  (B)  The  same  scene,  in  the  red-figured  style. 

26  Panel  Amphora.  Inv.  01.8037.  ( A ) Two  warriors  paying  draughts,  black- 
figured.  (B)  The  same  scene,  red-figured. 

These  two  amphorae  are  assigned  to  the  factory  of  the  potter  Andokides,  and 
are  decorated  by  the  same  painter,  called  by  Mr.  Beazley  the  “Andokides 
painter,”  Vases  in  America,  p.  3,  no.  Ia,  1 and  2.  They  resemble  one  another 
closely  in  their  main  proportions  and  the  forms  of  details,  but  evidently  do  not 
conform  to  the  same  geometrical  scheme.  The  chief  dimensions  are  as  follows : 


25  26 

Height 0.5325  m.  0.5536  m. 

Diameter 0.3465  0.35 

Diameter  of  lip 0.24  0.229 

Diameter  of  foot 0.203  0.204 


C 60  ] 


AMPHORA 


The  chief  differences  are  in  the  height  and  the  diameter  of  the  lip.  No.  25 
fits  fairly  accurately  the  rectangle  1.528.  For  no.  26  Mr.  Hambidge  proposed  the 
ratio  1.5801.  But  attempts  to  express  the  proportions  of  details  in  terms  of  these 
rectangles  have  been  unsuccessful. 

In  the  drawing  of  no.  26  the  enclosing  rectangle  has  been  given  the  ratio 
1.5773  (cf.  the  red-figured  amphora,  no.  42).  This  is  made  up  of  a square  and  a 
a/3  rectangle  (0.351  m.X  1.5773  = 0.5536  m.).  The  \/S  rectangle  is  placed  at  the 
bottom,  and  its  division  into  three  a/3  rectangles  accounts  for  the  diameter  of  the 
bottom  of  the  body  (ratio:  .333).  The  diameters  of  the  top  of  the  neck  and  the 
upper  member  of  the  foot  are  equal  to  half  the  total  width  (.500).  The  diameter 
of  the  foot  is  obtained  from  the  intersection  of  the  diagonal  of  one  of  the  three 
small  a/3  rectangles  with  the  diagonal  of  a square  applied  to  it  (.5118).  If  a y/S 
rectangle  is  applied  at  the  top  of  the  containing  shape,  the  diagonal  of  its  recipro- 
cal cuts  the  diagonal  of  the  containing  shape  so  as  to  determine  the  diameter  of 
the  lip  (.6512). 


C 61  ] 


GEOMETRY  OF  GREEK  VASES 


RED-FIGURED  PERIOD 

The  collection  includes  red-figured  amphorae  of  five  types.  The  first  twelve 
examples,  nos.  27-38,  usually  known  as  “ Nolan  amphorae,”  succeed  the  neck 
amphorae  of  the  black-figured  period.  Nos.  39-41  are  derived  from  the  Pana- 
thenaic  prize  amphorae  (cf.  nos.  20,  21).  No.  42  resembles  the  black-figured  panel 
amphorae  in  that  it  has  no  division  between  neck  and  shoulder.  No.  43  is.  a 
loutrophoros  — a vessel  used  in  the  Athenian  marriage  ceremonies.  No.  44  is  a 
midget  amphora  with  pointed  base,  of  the  late  fifth  century. 

A.  Nolan  Amphora 

Twelve  examples  in  the  collection  are  susceptible  of  accurate  measurement. 
Ten  are  of  the  normal  “Nolan”  type;  the  other  two,  nos.  29,  36,  differ  in  the 
profiles  of  neck  and  foot,  and  are  furnished  with  cylindrical  excrescences,  or  lugs, 
placed  horizontally  below  the  handles.  The  main  dimensions  of  the  amphorae, 
together  with  a description  and  bibliography  sufficient  to  identify  them,  are 
given  below  the  drawings.  The  arithmetical  ratios  are  recorded  in  the  table 
on  page  75. 


[ 62  ] 


AMPHORA 


27  Inv.  03.816.  Beazley,  V.  A.,  p.  69,  no.  I,  1.  (A)  Eos  and  Tithonos.  (B) 
Youth  running  with  lyre.  “ Painter  of  the  Boston  Tithonos.”  Width  (diameter  of 
body) , 0. 185  m.  The  missing  foot  is  restored  in  plaster,  so  that  the  ratio  cannot  be 
determined  with  certainty.  A rectangle  with  the  ratio  1.6545  allows  for  a foot  of 
normal  thickness.  The  height  has  therefore  been  assumed  to  be  0.306  m. 


[ 63  ] 


GEOMETRY  OF  GREEK  VASES 


28  Inv.  10.184.  Beazley,  V.  A.,  p.  115.  Hambidge,  p.  78.  (A)  Zeus  and 
Ganymede.  (B)  Youth  running  with  leg  of  meat.  “ Pan  Painter.” 

Height,  0.255  m.  Width  (diameter  of  body),  0.151  m.  The  ratio  is  1.691, 

or  1 4 

1 + V5 


[ 64  ] 


AMPHORA 


29  Inv.  76.46.  Beazley,  V.  A.,  p.  74,  no.  I,  1.  Seilens,  one  carrying  his  old 
father  on  his  back. 

Height,  .3205  m.  Width,  including  lugs,  0.185  m.  These  dimensions  give  the 
ratio  1.732,  or  \/3. 


[ 65  ] 


GEOMETRY  OF  GREEK  VASES 


30  Inv.  01.8109.  Beazley,  V.  A.,  p.  115.  Hambidge,  p.  61.  ( A ) Trainer  and 
athlete.  ( B ) Youth  drawing  along  the  hind  quarters  of  an  ox.  “ Pan  Painter.” 
Height,  0.307  m.  Width  (diameter  of  body),  0.179  m.  Ratio:  1.732  = \/3. 


[ 66  ] 


AMPHORA 


31  Inv.  13.188.  Beazley,  V.  A.,  p.  69.  Hambidge,  p.  60.  ( A ) Hephaistos 
polishing  the  shield  of  Achilles  in  the  presence  of  Thetis.  (B)  Nike  running. 
“ Dutuit  Painter.’7 

Height,  0.3425.  Width  (diameter  of  body),  0.196  m.  The  ratio  has  been  as- 
sumed to  be  1.732,  or  \/3.  To  obtain  this  the  height  has  been  reduced  1 mm. 
and  the  width  increased  1 mm.  The  actual  dimensions  give  more  nearly  the  ratio 
1.750.  But  if  that  rectangle  is  used,  the  details  work  out  less  accurately. 


[67] 


GEOMETRY  OF  GREEK  VASES 


32  Inv.  76.43.  Beazley,  V.  A.,  p.  166.  (A)  Woman  and  youth.  ( B ) Youth, 
“Painter  of  the  Dwarf  Pelike.” 

Height,  0.316  m.  Width  (handles),  0.18  m.  Ratio,  1.750. 


[68] 


AMPHORA 


33  Inv.  95.20.  Beazley,  V.  A.,  p.  138.  (A)  Nike.  (B)  Woman. 
Height,  0.331  m.  Width  (bowl  and  handles),  0.184  m.  R,atio,  1.809. 


[ 69  ] 


GEOMETRY  OF  GREEK  VASES 


34  Inv.  01.16.  Beazley,  V.  A.,  p.  168,  no.  II,  6.  Hambidge,  p.  80.  (A)  Woman 
with  oinochoe  and  man  with  phiale  at  altar.  (B)  Woman.  “ Painter  of  the  Bos- 
ton Phial.e.” 

Height,  0.332  m.  Width  (handles),  0.182  m.  Ratio,  1.809.  If  the  diameter 
of  the  bowl  be  taken  as  the  width,  the  ratio  is  2.000,  or  y/4,  the  diameter  of  the 
bowl  0.166  m.,  being  exactly  half  the  height. 


[ 70  ] 


AMPHORA 


35  Inv.  01.18.  Beazley,  V.  A.,  p.  138.  Hambidge,  p.  80.  (A)  Woman  pour- 
ing wine  for  a young  soldier.  (j B)  Youth.  “ Painter  of  the  Ethiop  Pelike.” 
Height,  0.3345  m.  Width  (handles),  0.1825  m.  Ratio,  1.809. 


[ 71  ] 


GEOMETRY  OF  GREEK  VASES 


36  Inv.  01.8028.  Beazley,  V.  A.,  p.  110,  no.  I,  2.  (4)  Maenad  dancing  and 
seilen  playing  the  flute.  (B)  Seilen  with  castanets.  “ Briseis  Painter.” 

Height,  0.2838  m.  Width  (diameter  of  bowl),  0.157  m.  These  dimensions  give 
the  ratio  1.809,  on  which  the  analysis  is  based.  If  the  distance  apart  of  the  lugs 
(0.168  m.)  were  taken  as  the  width,  the  ratio  would  be  1.691,  as  in  no.  28. 


C 72  ] 


AMPHORA 


37  Inv.  90.157.  Robinson,  Cat.  422  (A)  Two  women  sacrificing  at  an  altar. 
( B ) Youth. 

Height,  0.3455  m.  Width  (diameter  of  bowl),  0.187  m.  Ratio,  1.854. 


[73] 


GEOMETRY  OF  GREEK  VASES 


38  Inv.  06.2447.  Beazley,  V.  A.,  p.  164.  (A)  Oedipus  and  the  Sphinx.  ( B ) 
Youth.  “ Achilles  Painter.” 

Height,  0.332  m.  Width  (handles),  0.174  m.  Ratio,  1.9045. 

The  proportions  of  these  amphorae  are  expressed  arithmetically  in  the  follow- 
ing table,  the  greatest  width  being  regarded  as  unity: 


C 74  ] 


AMPHORA 


Number 

Width 

Height 

Height  to 
shoulder 

Height  of 
neck  and 
lip 

Diameter 
of  lip 

Smallest 
diameter 
of  neck 

Diameter 
of  shoulder 

Diameter 
of  bottom 
of  body 

Diameter 
of  foot 

27 

1.000 

1.6545 

1.1545 

.500 

.809 

.472 

.528 

.264 

? 

28 

1.000 

1.691 

1.2242 

.4668 

.764 

.4472 

.516 

.2764 

.516 

29 

1.000 

1.732 

1.2028 

.5292 

.7182 

.4444 

.500 

.2594 

.4811 

30 

1.000 

1.732 

1.251 

.4811 

.750 

.436 

.500 

.250 

.500 

31 

1.000 

1.732 

1.2268 

.5052 

.750 

.436 

.500 

.2594 

.4811 

32 

1.000 

1.750 

1.250 

.500 

.8666 

.4444 

.500 

.275 

.4444 

33 

1.000 

1.809 

1.2696 

.5394 

.809 

.4472 

.500 

.250 

.500 

34 

1.000 

1.809 

1.2696 

.5394 

.8333 

.4472 

.500 

.236 

.472 

35 

1.000 

1.809 

1.309 

.500 

.809 

.4472 

.500 

.236 

.500 

36 

1.000 

1.809 

1.2696 

.5394 

.7236 

.4472 

.5394 

.250 

.500 

37 

1.000 

1.854 

1.354 

.500 

.764 

.4472 

.528 

.264 

.472 

38 

1.000 

1.9045 

1.2865 

.618 

.854 

.4472 

.528 

.2236 

.4472 

The  twelve  amphorae  are  of  seven  different  shapes,  six  of  which  belong  to  the 
dynamic  class.  The  shape  1.809  occurs  four  times,  the  -\/3  rectangle  three  times, 
the  others  only  once.  The  simple  geometrical  divisions  of  these  seven  rectangles 
are  made  clear  by  the  small  diagrams  at  the  foot  of  this  page.  The  analysis  of 
details  in  the  larger  drawings  will  in  most  cases  be  intelligible,  if  examined  in 
connection  with  the  small  diagrams  and  the  table  of  ratios. 

Considered  from  the  point  of  view  of  chronology,  it  is  noteworthy  that  the 
Nolan  amphorae  become  slenderer  as  time  goes  on.  The  first  five  examples  belong 
to  the  ripe  archaic  period,  the  last  seven,  with  one  exception  (no.  36)  to  the  late 
archaic  or  early  free  style.  It  is  also  noticeable  that  the  elongation  is  all  in  the 
upper  part  of  the  vase  — the  shoulder  and  neck.  In  every  example  except  no.  1, 
the  body  attains  its  greatest  diameter  at  approximately  the  level  of  the  top  of  a 
square  inscribed  on  the  base  of  the  rectangle. 

Another  point  brought  out  by  the  table  is  the  very  frequent  occurrence  of  the 
simple  ratio  .500  (one-half  of  the  width).  It  is  found  four  times  as  the  height  of 
neck  and  lip,  seven  tunes  as  the  diameter  of  the  shoulder,  four  times  as  the  diam- 
eter of  the  foot.  The  diameter  of  the  bottom  of  the  body  is  .250,  or  one-quarter  of 
the  width,  in  three  cases. 

In  seven  examples  the  diameter  of  the  neck  is  to  the  total  width  as  the  height 
of  a v/5  rectangle  is  to  its  length  (.4472:  1.000)  and  only  in  one  case  (no.  27)  is 
there  a marked  departure  from  that  proportion. 


[ 75  ] 


GEOMETRY  OF  GREEK  VASES 


B.  Amphoka  of  Panathenaic  Shape 


39  Inv.  95.19.  Beazley,  V.  A.,  p.  25. 

(A,  B ) Figure  of  Athena. 

“Nikoxenos 

Painter.” 

Height,  0.435  m.  Diameter, 
The  proportions  are: 

0.284  m. 

Ratio,  1.528. 

Height 

1.528 

Diameter  of  lip 

5669 

Height  to  shoulder 

1.219 

Diameter  of  shoulder 

382 

Height  of  neck  and  lip 

.309 

Diameter  of  bottom  of  body 

292 

Diameter 

1.000 

Diameter  of  foot 

472 

The  proportions  of  the  foot  are  obtained  by  simple  subdivisions  of  the  rectangle 
.528,  as  shown  in  diagram  a. 

The  height  of  the  neck  and  lip  and  the  diameter  at  the  shoulder  and  the  top  of 
the  neck  are  obtained  from  the  rectangle  .309,  or  two  whirling  square  rectangles. 
One  of  these  is  shown  in  the  upper  left-hand  corner  of  the  drawing. 

The  diameter  of  the  lip  is  most  simply  expressed  in  terms  of  the  rectangle 
.3504  ( = 2.854),  as  shown  in  diagram  b.  Only  the  right-hand  half  of  this  rec- 
tangle is  shown  in  the  drawing. 


[ 76  ] 


AMPHORA 


[ 77  ] 


GEOMETRY  OF  GREEK  VASES 

40  Inv.  96.719.  Beazl ey,V.A.,  p.  122,  no.  II,  4.  (A)  Citharode  and  Athena.  (B) 
Hermes  and  Poseidon.  "Nausikaa  Painter.”  Height,  0.512m.  Diameter,  0.323  m. 

The  ratio  is  1.5854.  In  the 
drawing  a square  is  applied  at  the 
top  of  the  containing  rectangle. 
The  excess  area  at  the  bottom 
(.5854=1.708)  accounts  for  the 
proportions  of  the  foot.  The 
proportions  of  the  lip,  neck,  and 
shoulder  are  expressed  simply 
and  accurately  in  terms  of  a 
whirling  square  rectangle  applied 
at  the  top.  The  ratios  are: 


Height 1.5854 

Height  to  shoulder 1.2034 

Height  of  neck  and  lip 382 

Diameter 1.000 

Diameter  of  lip 5528 

Diameter  of  shoulder 382 

Diameter  of  bottom  of  body  .2764 
Diameter  of  foot 4146 


The  ratio  1.5854  is  very  close 
to  another  ratio,  1.5858,  which 
belongs  to  the  y/2  system.  The 
rectangle  .5858  (=  1.7071)  ac- 
counts satisfactorily  for  the  diam- 
eter of  the  foot,  the  ratio  being 
.4142  (1.000 -.5858),  or  V2-1. 
But  the  ratio  .4146  (1.000  — .5854) 
is  hardly  distinguishable  in  a 
drawing  from  .4142.  The  rectangle  .5854  also  accounts  simply  for  the  diameter 
of  the  bottom  of  the  body.  All  the  proportions  can  thus  be  expressed  in  terms 
of  the  whirling  square  rectangle,  whereas  only  one  of  the  details  fits  a simple 
ratio  derived  from  \/2. 

41  Inv.  10.178.  Beazley,  V.  A.,  p.  42.  Hambidge,  p.  130.  (A)  Athlete  with 
presents.  (B)  Youth  offering  wreath.  “ Kleophrades  Painter.”  Height,  0.454  m. 
Diameter,  0.2815  m.  Ratio,  1.618,  or  a whirling  square  rectangle. 

The  proportions  are: 


Height 

. ...  1.618 

Diameter 

. 1.000 

Height  to  shoulder 

, ...  1.309- 

Diameter  of  lip 

. .5528 

Height  of  neck  and  lip 

309+ 

Diameter  of  shoulder 

. .3416 

Height  of  lip 

1382 

Diameter  of  bottom  of  body . . . 

. .236+ 

Height  of  neck 

1708 

Diameter  of  foot 

. .427 

[ 78  ] 


AMPHORA 


There  is  an  inaccuracy  of  0.00175  m.  in  the  height  of  the  shoulder.  It  is  note- 
worthy that  the  lip  is  inclosed  in  four  squares,  and  the  neck  in  two  squares. 


[ 79  ] 


GEOMETRY  OF  GREEK  VASES 


C.  Amphora  without  Division  between  Neck  and  Shoulder 


42  Inv.  98.882.  Beazley,  V.  A.,  p.  59,  no.  I,  4.  ( A ) Seilen  holding  his  son  on 
his  shoulders.  (B)  Seilen  with  ph alios.  “ Flying-angel  Painter.” 

Height,  0.4085  m.  Diam- 
eter, 0.2604  m. 

This  amphora  is  not  a 
masterpiece  of  pottery;  and 
the  dimensions  given  are 
averages  obtained  from  sev- 
eral measurements  which 
vary  more  than  0.005  m.  in 
the  case  of  the  height,  and 
0.003  m.  in  the  case  of  the 
diameter.  If  the  height  is 
taken  as  0.41  m.  and  the  di- 
ameter as  0.26  m.,  the  result- 
ing ratio  is  1.5773.  .5773  is 
the  reciprocal  of  1.732,  or\/3. 
The  containing  rectangle  is 
therefore  close  to  the  area 
composed  of  a square  and  a 
s/3  rectangle.  The  propor- 
tions of  details  are  expressible 
in  simple  ratios,  three  of 
which  occur  in  “static  sym- 
metry.” 


The  ratios  are  as  follows : 

Height 1. 57734- 

Height  omitting  lip . . 1.500 

Height  of  foot 0962  = .5773  -r-  6 

Diameter 1.000 


Diameter  of  lip 57734- 

Diameter  of  bottom  of  body 333 

Diameter  of  foot 5004- 


[ 80  ] 


AMPHORA 


D.  Lotjtrophoros 

43  Inv.  03.802.  Annual  Report,  1903, 
p.  71,  no.  62.  ( A ) Bridal  procession.  (B) 
Parting  scene.  Dimensions  given  in  Annual 
Report:  Height,  0.753  m. ; diameter  of  lip, 
0.253  m.  In  our  drawing  the  diameter  of 
the  lip  is  taken  as  0.25  m.  The  original 
height  is  not  obtainable  exactly,  since  there 
is  some  restoration  at  the  bottom  of  the 
body.  It  has  been  assumed  to  be  0.75  m., 
giving  a rectangle  made  up  of  three  squares, 
or  \/9.  It  is  probable  that  the  width  ought 
to  have  been  taken  as  somewhere  between 
0.250  m.  and  0.253  m.  This  would  bring 
the  height  of  the  shoulder  more  exactly  in 
conformity  with  the  scheme. 

Using  the  diameter  of  the  lip  as  unity, 
the  ratios  are: 


Height 3.000? 

Height  to  shoulder 1.646? 

Height  of  neck  and  lip 1.354 

Height  of  lip 118 

Height  of  neck 1.236 

Diameter  of  lip 1 .000 

Diameter  at  j unction  of  neck  and  lip  .447 2 

Diameter  of  neck 236  ± 

Diameter  of  shoulder 333 

Greatest  diameter  of  body 708 

Smallest  diameter  of  body 191 

Diameter  of  foot 764 


43 


[ 81  ] 


GEOMETRY  OF  GREEK  VASES 


E.  Midget  Pointed  Amphora 


44  Inv.  00.355.  Beazley,  V.  A.,  p.  180.  (A)  Eros.  (B)  Seated  woman.  Free 
style. 

Height,  0.191  m.  Diameter,  0.087  m.  The  rectangle  has  the  ratio  2.2071.  In 

the  drawing  this  has  been  divided  into  the  three  rectangles  .7071  .500  and 

1.000.  The  shoulder  is  nearly  at  the  height  1.500.  The  diameter  of  lip  and 
shoulder  may  be  expressed  by  squares  applied  at  either  end  of  the  .7071  rec- 
tangle. On  account  of  the  small  size  of  the  vase  no  further  analysis  is  attempted. 


[ 82  ] 


PELIKE 


PELIKE 

The  collection  contains  one  example  of  this  type  of  vase  with  black-figured 
decoration,  but  it  is  of  such  inferior  workmanship  that  no  drawing  has  been  made. 
Its  ratio  is  apparently  If.  Six  red-figured  examples  are  sufficiently  well-pre- 
served to  justify  an  examination  of  their  proportions.  The  ratios  so  far  as  they 
have  been  determined,  are  as  follows : 


Height 

Width 

Diameter 
of  lip 

Smallest 
diameter 
of  neck 

Diameter 
of  bottom  of 
body 

Diameter  of 
foot 

45 

1.2071 

1.000 

.4142 

.7071 

46 

1.236 

1.000 

.708 

.528 

.708 

.764 

47 

1.309 

1.000 

.7236 

.500 

.528 

.691 

48 

1.3587 

1.000 

.500 

.618 

.708 

49 

1.382 

1.000 

.809 

.528 

.618 

.764 

50 

1.4142 

1.000 

.500 

.5858 

[ 83  ] 


GEOMETRY  OF  GREEK  VASES 


45  Inv.  89.265.  (A)  Two  youths  conversing.  ( B ) Youth.  Free  style. 
Height,  0.1485  m.  Diameter,  0.123  m.  Diameter  of  lip,  0.09  m.  Diameter  of 
foot,  0.095  m. 

The  containing  rectangle  is  the  familiar  shape  1.2071  (.7071 + .500),  or 


+2+1 

2 


The  bottom  of  the  maeander  band  coincides  with  the  base  of  a square 


applied  at  the  top  of  the  rectangle.  The  excess  area  below  (.2071)  accounts  for 
the  diameter  of  the  bottom  of  the  body  and  the  smallest  diameter  of  the  neck. 
The  diameter  of  lip  and  foot  are  not  expressible  in  simple  ratios. 


[ 84  ] 


PELIKE 


46  Inv.  03.818.  ( A ) Boy  seated  on  a knoll,  tossing  a ball.  (B)  Nike  flying. 
Free  style. 

Height,  0.1665  m.  Diameter,  0.1345  m.  Diameter  of  lip,  0.096  m.  Diameter 
of  foot,  0.103  m. 

The  ratio  is  1.236,  or  two  whirling  square  rectangles,  and  the  proportions  can 
all  be  expressed  by  simple  subdivisions  of  these  rectangles.  The  lower  painted 
band  is  placed  at  the  same  height  in  the  lower  rectangle  as  the  upper  band  is  in  the 
upper  rectangle. 


[ 85  ] 


GEOMETRY  OF  GREEK  VASES 


47  Inv.  03.793.  Beazley,  V.  A.,  pi.  83,  no.  12.  Hambidge,  p.  92.  (A)  Soldier 
leaving  home.  ( B ) Youths.  “ Kleophon  Painter.” 

Height,  0.3585  m.  Width  (handles),  0.279  m.  Diameter  of  bowl,  0.275  m. 
Diameter  of  lip,  0.20  m.  Diameter  of  foot,  0.191  m. 

If  the  height  be  assumed  to  be  0.359  m.  and  the  diameter  0.274276  m.,  the 
ratio  is  1.309.  This  rectangle  has  been  used  in  the  analysis;  the  slight  projection  of 
the  handles  beyond  it  has  not  been  considered. 

The  rectangle  1.309  can  be  analyzed  in  various  ways.  In  the  present  example 
the  division  of  it  into  the  rectangles  .618  and  .691  seems  to  produce  the  most 
significant  results.  The  ratios  are: 

Height 1.309  Smallest  diameter  of  neck 500 

Diameter 1 .000  Diameter  of  bottom  of  body 528 

Diameter  of  lip 7236  Diameter  of  foot 691  — 

In  the  drawing  the  whirling  square  rectangle  has  been  placed  above  the  .691 
rectangle,  because  the  proportions  of  the  upper  part  of  the  vase  can  be  obtained 
from  simple  subdivisions  of  it.  The  diameter  of  the  bottom  of  the  body  is  also  ob- 
tained simply  from  the  whirling  square  rectangle,  as  is  shown  in  the  drawing. 
The  diameter  of  the  foot  is  enclosed  in  a square  applied  at  the  centre  of  the  .691 
rectangle.  It  is  noteworthy  that  .691  is  the  reciprocal  of  1.4472,  i.  e.,  a square  plus 
a \/5  rectangle,  the  shape  that  underlies  the  proportions  of  the  Parthenon.  The 
ratio  .7236  is  closely  connected  with  it,  since  it  is  the  reciprocal  of  1.382,  or 
.691 X 2.  It  is  also  noteworthy  that  1.000  — .691  = .309. 


[ 86  ] 


PELIKE 


[ 87] 


GEOMETRY  OF  GREEK  VASES 


48  Inv.  98.883.  Beazley,  V.  A.,  p.  169,  no.  iv,  21.  (A)  Actors  dressing  as 
women.  ( B ) Man.  “ Painter  of  the  Boston  Phiale.” 

Height,  0.2405  m.  Diameter,  0.177  m.  Diameter  of  lip,  0.138  m.  Diameter  of 
foot,  0.1255  m. 

No  satisfactory  analysis  of  this  vase  has  been  found.  The  ratio  1.3587  (recip- 
rocal .736)  is  uncommon.  The  proportions  given  in  the  table  have  been  worked 
out  arithmetically.  The  geometrical  analysis  is  omitted. 


[ 88  ] 


PELIKE 


49  Inv.  76.45.  Beazley,  V.  A.,  p.  166.  (A)  Youth  walking,  followed  by  his 
dwarf  servant  and  dog.  (B)  Youth.  “ Dwarf  Pelike  Painter.” 

Height,  0.2407  m.  Width  (handles),  0.19  m.  Diameter  of  bowl,  0.175  m. 
Diameter  of  lip,  0.141  m.  Diameter  of  foot,  0.135  m. 

The  enclosing  rectangle  has  the  ratio  1.2677,  the  reciprocal  of  which  is  .7888, 
i.  e.  (.4472X4)  — 1.000.  If  the  diameter  of  the  bowl  is  taken  as  unity,  the  ratio  is 
1.382.  This  familiar  rectangle  has  been  used  in  the  analysis.  The  proportions  of 
details  are  simply  obtained  by  applying  whirling  square  rectangles  at  the  top  and 
bottom,  and  subdividing  their  reciprocals. 


[ 89  ] 


GEOMETRY  OF  GREEK  VASES 


50  Inv.  20.187.  Beazley,  V.  A.,  p.  161,  no.  1, 1.  ( A ) Nike  setting  up  a trophy. 
(B)  Youth,  early  free  style.  “ Painter  of  the  Deepdene  Trophy.” 

Height,  0.2545  m.  Width  (handles),  0.181m.  Diameter  of  bowl,  0.179m. 
Diameter  of  lip,  0.132  m.  Diameter  of  foot,  0.13  m. 

If  the  width  be  taken  as  0.18  m.,  and  this  be  multiplied  by  y/2  (1.4142),  the 
result  is  0.254556  m. 

The  drawing  shows  that  the  smallest  diameter  of  the  neck  is  close  to  half  the 
total  width  (.500).  The  diameter  of  the  bottom  of  the  bowl  is  determined  by  the 
intersection  of  the  diagonal  of  the  overall  y/2  rectangle  with  the  diagonal  of  a 
square  inscribed  on  half  the  base.  The  ratio  is  .5858,  which  is  the  reciprocal  of 
1.7071.  No  simple  ratios  have  been  found  for  the  diameters  of  the  lip  and  the 
foot. 


C 90  ] 


STAMNOS 


STAMNOS 

In  the  red-figured  period  the  stamnos,  a jar  with  a short  neck,  wide  mouth  and 
horizontal  handles,  seems  to  have  taken  the  place  of  the  large  amphora  as  a con- 
tainer for  wine.  Five  examples  in  the  Museum  have  been  selected  for  publication, 
the  first  four  of  which  are  of  the  normal  type.  A stamnos  in  Professor  Hoppin’s 
collection  is  added  because  of  the  interest  of  its  geometrical  scheme.  Two  ex- 
amples in  the  Museum  (Inv.  95.21  = Beazley,  V.  A.,  p.  172;  Inv.  10.177  = Beazley, 
V.  A.,  p.  102)  have  been  omitted  because  they  are  misshapen,  and  four  others  be- 
cause they  are  incomplete  *or  of  poor  workmanship. 


I 


[ 91  ] 


GEOMETRY  OF  GREEK  VASES 


51  Stamnos,  with  cover.  Inv.  90.155.  Beazley,  V.  A.,  p.  154.  Women  bring- 
ing offerings  to  a rustic  idol  of  Dionysos.  “ Villa  Giulia  Painter.” 

Height,  with  cover,  0.48  m.  Width  (handle  to  handle),  0.431  m.  Height  with- 
out cover,  0.399  m.  Diameter  of  bowl,  0.335  m. 

•\/5 

The  stamnos  with  its  cover  fits  the  shape  1.118  (.8944),  i.e.,  — , or  two  \/5 

A 

rectangles  placed  vertically  side  by  side  (cf.  diagram  A).  Without  its  cover  the 
vase  is  contained  in  the  area,  .927,  i.  e.,  .618+.309,  or  a whirling  square  rectangle 
plus  two  whirling  square  rectangles  (cf.  diagram  B).  The  vase  up  to  the  shoulder 
is  contained  in  the  area  .809,  or  two  vertical  whirling  square  rectangles  (diagram 
A).  The  portion  above  the  shoulder  is  enclosed  in  two  horizontal  whirling  square 
rectangles,  the  ratio  being  .309  (diagram  A).  The  ratios  .809  and  .309  added  give 
the  over-all  ratio,  1.118.  If  the  handles  as  well  as  the  cover  are  omitted,  the 


enclosing  rectangle  has  the  ratio  1.191,  i.  e.,  .809  + . 382  (cf.  diagram  C).  The  rela- 
tion of  the  .809  rectangle  in  diagram  A to  that  in  diagram  C is  shown  by  the 
diagonals  in  the  latter.  If  the  cover  is  considered  separately,  it  is  seen  to  be  en- 
closed in  the  rectangle  2.8944,  which  is  made  up  of  two  squares  plus  two  \/5 
rectangles  (diagram  C). 

The  proportions  of  details  can  be  studied  with  the  help  of  the  large  drawing 
and  the  following  table  of  ratios. 


Height,  with  cover 1.118 

Height,  without  cover 927 

Height  of  cover 191 

Height  to  shoulder 809 

Height  of  neck  and  lip 118 

Width 1.000 


Diameter  of  bowl 7783 

Diameter  of  lip 58944 

Diameter  of  cover 5528 

Diameter  of  shoulder 4809 

Diameter  of  foot 3455 


[ 92  ] 


STAMNOS 


[ 93  ] 


# 


GEOMETRY  OF  GREEK  VASES 

52  Inv.  00.342.  Beazley,  V.  A.,  p.  145,  no.  IV,  18.  (A)  Dionysos  and  a giant. 
OB)  Seilen  driving  a pair  of  seilens.  “ Altamura  Painter.” 

Height,  0.382  m.  Width,  including  handles,  0.404  m.  Height  to  shoulder, 
0.329  m.  Diameter  of  bowl,  0.392  m.  Diameter  of  lip,  0.233  m.  Diameter  of  foot, 

0. 191  m. 

In  the  drawing  the  stamnos  has  been  enclosed  in  the  rectangle  .9472  (1.0557), 

1.  e.,  a \/5  rectangle  placed  above  two  squares.  The  vase  without  the  neck  and 
handles  is  enclosed  in  a square.  The  lower  margin  of  the  maeander  band  encir- 
cling the  vase  coincides  with  the  base  of  a whirling  square  rectangle  applied  to  the 
upper  part  of  the  over-all  shape.  The  intersection  of  the  diagonal  of  this  whirling 
square  rectangle  with  the  diagonal  of  a quarter  of  the  V 5 rectangle  determines 
the  height  of  the  shoulder  and  the  diameter  of  the  lip.  The  diameter  of  the  foot 
equals  half  the  height  of  the  stamnos. 

Though  in  some  respects  apparently  simple,  the  geometry  of  this  vase  remains 
a puzzle  to  the  writer.  The  correspondence  of  several  of  the  dimensions  measured 
according  to  the  metric  system  with  simple  ratios  (.382,  .4045,  .3292,  .191)  in  this 
case  seems  to  increase  rather  than  diminish  the  difficulty  of  explaining  the  pro- 
portions. The  analysis  is  to  be  regarded  as  tentative,  and  the  table  of  ratios  is 
omitted. 


C 94  2 


STAMNOS 


* 


C 95  ] 


GEOMETRY  OF  GREEK  VASES 


53  Stamnos,  with  cover.  Inv.  01.8083.  Beazley,  V.  A.,  p.  155,  no.  6,  Maenads. 
“Painter  of  the  Chicago  Stamnos.” 


54  Stamnos,  with  cover.  Inv.  01.8082.  Beazley,  V.  A.,  p.  155,  no.  7,  Komos. 
“Painter  of  the  Chicago  Stamnos.” 

These  two  vases  were  decorated  by  the  same  painter.  They  are  almost  iden- 
tical in  size,  and  are  evidently  from  the  same  factory.  The  chief  dimensions  are : 


Height 
with  cover 

Height 
without  cover 

Width 

Diameter  of 
bowl 

Diameter  of 
lip 

Diameter  of 
foot 

53 

.3755 

.327 

.374 

.281 

.213 

.136 

54 

.377 

.331 

.373 

.278 

.216 

.1295 

[ 96  ] 


STAMNOS 


The  over-all  shape  of  no.  53  is  a square,  and  study  of  its  details  shows  that  it 
belongs  to  the  “static  ” class.  With  the  lid  removed,  the  shape  is  f (.875). 
If  the  lid,  the  lip  and  neck,  and  the  handles  are  removed,  the  shape  is  again  a 
square.  The  proportions  of  details  cannot  in  every  case  be  expressed  accurately 
in  simple  terms  of  the  over-all  shape. 

The  stamnos  no.  54  does  not  fit  accurately  any  scheme  either  “ static  ” or 
“dynamic,”  so  far  as  I have  been  able  to  determine.  It  approaches  fairly  closely 
to  the  scheme  of  no.  53  as  is  shown  in  the  drawing. 


[ 97  ] 


GEOMETRY  OF  GREEK  VASES 


55  Stamnos,  with  flat  cover.  Inv.  00.349.  (A)  Theseus  abandoning  Ariadne. 
( B ) Bellerophon  with  Pegasos.  Free  style. 

Height,  with  cover,  0.3185  m.  Width  (handle  to  handle),  0.32  m.  Height 
without  cover,  0.2965  m.  Diameter  of  bowl,  0.259  m. 

The  enclosing  area  is  very  nearly  a square  (diagram  A) . Without  its  cover  the 
stamnos  is  contained  in  a rectangle  with  the  ratio  .927,  i.  e.,  .618+. 309  (diagram 
B).  If  the  cover  is  included  and  the  handles  are  omitted,  the  area  is  composed  of 
two  whirling  square  rectangles  (diagram  C) . The  shoulder  is  at  half  the  height  of 
the  small  whirling  square  rectangles  in  diagram  B. 


The  ratios  are : 

Height,  with  cover . . . 
Height,  without  cover 
Height,  to  shoulder . . 

Width 

Diameter  of  bowl .... 

Diameter  of  lip 

Diameter  of  foot 


1.000 

.927  = .618  + .309 
.7725  = .618  + .1545 


1.000 

.809 

.691 


.500  (inaccurate) 


[ 98  ] 


STAMNOS 


[ 99  ] 


GEOMETRY  OF  GREEK  VASES 


■ .618- 


56 


>♦045 


.809 


56  Stamnos  in  the  collection  of  Professor  J.  C.  Hoppin  at  Pomfret,  Conn. 

Height,  0.450  m.  Width  (handle  to  handle),  0.443  m.  Diameter  of  bowl, 
0.365  m.  Diameter  of  lip,  0.271  m.  Diameter  of  foot,  0.1595  m. 

The  two  accompanying  drawings  were  made  from  measurements  and  a full- 
size  elevation  furnished  by  Professor  Hoppin.  His  drawing  did  not  include  the 
handles;  these  have  been  added  in  the  diagrams  in  dotted  lines  to  suggest  ap- 
proximately their  appearance. 

The  enclosing  area  is  a rectangle  with  the  ratio  1.0225,  i.  e.,  .618+. 1545,  or  a 
horizontal  whirling  square  rectangle  with  four  vertical  whirling  square  rectangles 
placed  above  it.  The  height  of  the  shoulder,  and  the  diameters  of  lip,  shoulder, 
and  bottom  of  body  are  obtained  from  simple  subdivisions  of  the  containing 
shape.  The  stamnos  without  its  handles  is  contained  in  the  familiar  rectangle 
.809  or  1.236,  i.  e.,  two  horizontal  whirling  square  rectangles.  The  diagonals  in  the 
lower  half  of  the  second  drawing  show  the  relation  of  the  lower  of  these  two 
whirling  square  rectangles  to  the  whirling  square  rectangle  of  the  over-all  shape. 
The  diameter  of  the  neck  is  seen  to  be  one-half  the  diameter  of  the  body. 

The  ratios  are : 


Height 

1.0225 

Diameter  of  lip 

. .618 

Height  to  shoulder . . 

.868  = 

.618  + .250 

Smallest  diameter  of  neck .... 

. .4136 

Height  of  neck  and  lip 

.1545  = 

.618  -i-  4 

Diameter  of  shoulder 

. .427 

Width 

1.000 

Diameter  of  bottom  of  body . . 

. .236 

Diameter  of  bowl . . . 

.8272 

Diameter  of  foot 

. .354 

[ 100  ] 


HYDRIA-KALPIS 


HYDRIA-KALPIS 

Two  types  of  water  jar  are  distinguished  by  these  names.  The  hydria  is  a jar 
with  two  horizontal  handles  at  the  sides  for  lifting  and  a vertical  handle  at  the 
back  for  pouring.  The  junction  between  neck  and  shoulder  is  definitely  marked, 
and,  except  in  no.  57,  the  sloping  shoulder  is  also  definitely  distinguished  from  the 
body.  This  type  lasted  down  into  the  ripe  archaic  red-figured  period.  In  the  later 
red-figured  period  its  place  was  taken  by  the  kalpis,  a jar  broader  in  proportion  to 
its  height,  and  differing  also  in  the  shapes  of  its  handles  and  foot,  and  in  the 
absence  of  lines  of  demarcation  between  neck,  shoulder  and  body. 

The  geometrical  schemes  of  the  hydriae  nos.  58-60,  62,  and  of  the  kalpis  no.  65 
have  not  been  worked  out  satisfactorily.  The  variation  in  the  main  proportions 
is  shown  in  the  following  table: 


Ratio 

with  handles 

Ratio 

without  handles 

No.  57  B.  F.  Hydria 

1.0355 

1.178 

58  “ 

1.0787 

1.3455 

59  “ 

1.1056 

1.3455 

60  “ 

1.1708 

1.3618 

61  “ 

1.2071 

1.500 

62  “ 

1.3455 

1.500 

63  R.  F.  Hydria 

1.4472 

1.528 

64  R.  F.  Kalpis 

1.0652 

1.266 

65  “ * 

1.0787 

1.309 

66  “ 

1.0787 

1.333 

67  “ 

1.118 

1.382 

68  “ 

1.118 

1.309 

69  “ 

1.382 

1.618 

C 101  ] 


GEOMETRY  OF  GREEK  VASES 


BLACK-FIGURED  PERIOD 

57  Hydria.  Inv.  95.62.  Collection  Van  Branteghem , No.  6.  Hambidge, 
p.  55,  fig.  16;  photograph  opp.  p.  56.  On  shoulder,  bacchic  dance.  On  body, 
Dionysos,  with  seilens,  maenads,  and  a horseman  galloping.  The  vase  is  tech- 
nically a masterpiece.  The  only  blemish  is  the  unevenness  of  the  lip. 

Height  to  top  of  handle,  0.325  m.  Height  of  lip,  0.316  m.  Height  to  shoulder, 
0.238  m.  Width,  including  handles,  0.3365  m.  Diameter  of  body,  0.267  m. 
Diameter  of  lip,  0.191  m.  Diameter  of  foot,  0.1475  m. 

The  proportions  of  this  remarkable  vase  were  worked  out  by  Mr.  Hambidge 
on  the  basis  of  a drawing  made  by  the  writer  in  1918.  In  view  of  its  importance  a 
new  drawing  based  on  revised  measurements  is  here  published.  The  only  varia- 
tions are  a slight  reduction  of  the  diameter  of  the  bottom  of  the  neck  and  the 
enlargement  of  the  diameter  of  the  bottom  of  the  body  from  0.069  m.  to  0.073  m. 
The  scheme  of  proportions  remains  essentially  unchanged.  Though  complicated, 
it  is  obviously  in  terms  of  the  y/2  rectangle  throughout. 

If  the  height  to  the  top  of  the  handle  is  taken  as  unity,  the  width  has  the  ratio 
1.0355  (=  .2071X5).  The  amount  of  projection  of  each  handle  from  the  body 
is  .10355  (=.2071-7-2),  or  one-tenth  of  the  total  width. 

The  diameter  of  the  body  is  .8284  (=.2071X4).  This  is  the  reciprocal  of 
1.2071,  a familiar  ratio  represented  geometrically  by  half  a square  plus  a a/2 
rectangle.  If  the  lateral  handles  are  removed,  the  vase  fits  this  area  fairly  accu- 
rately. Cf.  diagram  A. 

The  diameter  of  the  lip  has  the  ratio  .5858,  which  is  the  reciprocal  of  1.7071, 
again  a familiar  ratio  represented  by  a square  and  a \/2  rectangle.  Cf.  diagram  B. 

The  diameter  of  the  foot  has  the  ratio  .4531,  the  reciprocal  of  which  is  2.2071 
expressed  geometrically  as  a square  plus  half  a square  plus  a a/2  rectangle.  The 
diameter  of  the  ring  at  the  bottom  of  the  body  equals  .2265,  or  half  the  diameter 
of  the  foot.  The  smallest  diameter  (just  above  the  ring)  is  .2071,  or  one-fifth  of 
the  total  width. 

It  is  noteworthy  that  the  main  horizontal  division  of  the  enclosing  rectangles 
in  diagrams  A and  B coincides  exactly  with  the  painted  band  immediately  below 
the  lower  zone  of  figures.  The  height  of  this  zone  exactly  equals  the  diameter  of 
the  ring  at  the  bottom  of  the  body  (.2265),  or  half  the  diameter  of  the  foot. 

The  interrelation  of  a number  of  these  proportions  is  shown  by  the  dotted  lines 
in  the  large  drawing. 

It  is  difficult  to  determine  the  height  of  the  lip  in  simple  terms  of  the  containing 
rectangle.  This  height  varies  between  0.314  m.  and  0.318  m.  If  the  height  is 
taken  as  0.31728  m.  the  vase,  omitting  the  projection  of  the  vertical  handle,  fits 

3V2 

the  rectangle  1.0606  (=  .3535  X 3),  or  — — . Cf.  diagram  C.  This  area  is  com- 
posed of  six  v/2  rectangles.  The  shoulder  is  seen  to  be  at  three-quarters  of  the 
total  height;  and  the  vase  up  to  the  shoulder  is  contained  in  a \/2  rectangle. 


[ 102  ] 


HYDRIA-KALPIS 


GEOMETRY  OF  GREEK  VASES 


58  Hydria.  Inv.  01.8125.  On  body,  four  women  carrying  water  jars  on  their 
heads.  On  shoulder,  a warrior  attacked  by  two  horsemen. 

Greatest  height,  0.36  m.  Width,  0.334  m.  Height  of  lip,  0.349  m.  Diameter  of 
body,  0.259  m. 

The  enclosing  rectangle  apparently  has  the  ratio  1.0787  ( = .927) . The  diameter 
of  the  lip  is  .618,  the  greatest  diameter  of  the  body,  .764,  the  diameter  of  the  bot- 
tom of  the  body,  .382,  the  diameter  of  the  foot,  .4045.  The  geometrical  analysis  is 
complicated,  and  is  therefore  omitted.  It  is  possible  that  the  ratio  is  1.073,  i.  e., 
.309  + . 764,  or  .691  + .382.  If  the  three  handles  are  omitted,  the  ratio  is  1.3455. 

59  Hydria.  Inv.  89.562.  On  body,  Herakles  mounting  a quadriga  in  the 
presence  of  Athena,  Apollo,  and  five  other  figures.  On  shoulder,  Theseus  killing 
the  Minotaur. 

Greatest  height,  0.4505  m.  Width,  0.407  m.  Height  of  lip,  0.43  m.  Greatest 
diameter  of  body,  0.32  m. 

The  ratio  is  apparently  1.1056.  This  is  .5528X2.  The  reciprocal  of  .5528  is 
1.809.  The  enclosing  rectangle  is  thus  composed  of  two  1.809  rectangles  placed 
horizontally  one  above  the  other.  The  shoulder  is  at  three-quarters  of  the  total 
height  (.8292).  The  height  of  neck,  lip  and  handle  is  thus  .2764,  or  .5528^-2.  The 
diameter  of  the  lip  is  .5669,  that  of  the  shoulder,  .3455,  that  of  the  body,  .7888, 
that  of  the  foot,  .3504.  The  hydria  without  the  handles  fits  the  rectangle  1.3455. 
The  geometrical  analysis  is  omitted. 

60  Hydria.  Inv.  99.522.  On  body,  Herakles  and  Triton.  On  shoulder,  three 
youths,  each  leading  a horse. 

Greatest  height,  0.409  m.  Width,  0.349  m.  Height  of  lip,  0.3955  m.  Greatest 
diameter  of  body,  0.29  m. 

The  ratio  is  apparently  1.1708  ( = .854).  Without  its  handles  the  hydria  fits  the 
rectangle  1.3618.  The  shoulder  is  at  three-quarters  of  the  height.  Most  of  the 

[ 104  ] 


HYDRIAKALPIS 


details  are  expressible  in  simple  ratios,  though  the  geometrical  analysis  has  again 
proved  complicated,  and  has  been  omitted.  The  ratios  are: 


Height 

Height  of  neck,  lip  and  handle 

Height  to  shoulder 

Width 

Diameter  of  lip 

Diameter  of  shoulder 

Diameter  of  body 

Diameter  of  bottom  of  body . . 
Diameter  of  foot 


1.1708 

.2927 

.8781 

1.000 

.5854 

.4146 

.8292 

.250 

.4472 


1.1708  -f-  4 
.2927  X 3 

1.1708  -f-  2 


[ 105  ] 


GEOMETRY  OF  GREEK  VASES 


61  Hydria.  Inv.  01.8060.  On  body,  eight  horses  being  watered  at  a trough. 
On  shoulder,  three  horsemen. 

Greatest  height,  0.456  m.  Width,  0.3775  m.  Height  of  lip,  0.436  m.  Greatest 
diameter  of  body,  0.291  m. 

The  hydria  fits  the  familiar  rectangle  1.2071.  Without  its  handles  it  is  enclosed 
in  the  rectangle  1.500.  The  height  of  neck,  lip  and  handle  is  one-third  of  the  total 
width  (.333),  shown  in  the  drawing  by  three  squares.  The  diameter  of  the 
shoulder  is  somewhat  less  than  this,  i.  e.,  it  nearly  coincides  with  the  central 
square.  The  diameter  of  the  lip  is  simply  obtained  by  describing  arcs  of  circles 
with  the  corners  of  the  central  square  as  centres  and  its  diagonal  as  radius.  The 
ratio  is  .6094.  The  diameter  of  the  foot  is  .4142. 


[ 106  ] 


HYDRIA-KALPIS 


RED-FIGURED  PERIOD 

62  Inv.  13.200.  Hydria.  Beazley,  V.  A.,  p.  52.  On  body,  carpenter  working 
on  chest  in  presence  of  Akrisios,  Eurydike,  and  Danae  with  Perseus  in  her  arms. 
On  shoulder,  three  youths  baiting  a bull. 

Greatest  height,  0.458  m.  Width,  0.34  m.  Height  of  lip,  0.4185  m.  Greatest 
diameter  of  body,  0.279  m.  Diameter  of  lip,  0.1915  m.  Diameter  of  foot,  0.132  m. 

The  over-all  ratio  is  1.3455.  If  the  handles  are  omitted  the  ratio  is  1.500.  Most 
of  the  details  can  be  accurately  expressed  in  simple  ratios,  but  the  geometrical 
analysis  is  too  complicated  to  be  easily  intelligible.  The  diagonal  lines  in  the 
drawing  are  diagonals  of  squares,  half  squares,  whirling  square  rectangles  and 
half  whirling  square  rectangles.  The  height  of  the  shoulder  has  not  been  deter- 
mined. The  diameter  of  the  lip  is  .5669,  that  of  the  shoulder,  .309,  that  of  the 
bottom  of  the  body,  .236,  that  of  the  foot,  .382.  The  projection  of  the  vertical 
handle  is  .118. 


C 107  ] 


GEOMETRY  OF  GREEK  VASES 


63  Hydria.  Inv.  98.878.  On  body,  a young  warrior  and  a woman  making  a 
libation.  On  shoulder,  two  lions  attacking  a bull.  Ripe  archaic  style. 

The  foot,  which  is  missing,  has  been  incorrectly  restored  in  plaster,  the  type  of 
foot  found  in  kalpides  having  been  used  in  place  of  the  simple  disc  proper  to 
hydriae  of  the  black-figured  type.  The  dimensions  are:  Height,  omitting  foot, 
0.565  m.  Height  from  bottom  of  neck  to  top  of  vertical  handle,  0.181  m.  Width, 
0.4045  m.  Diameter  of  bowl,  0.339  m.  Diameter  of  lip,  0.223  m. 

The  width,  0.4045  m. , divided  by  the  height  from  the  bottom  of  the  neck  to  the 
top  of  the  vertical  handle,  0.181  m.,  gives  the  ratio  2.236,  or  \/5.  All  the  details  of 
the  vase  above  the  shoulder  can  be  easily  obtained  from  subdivisions  of  the  -y/5 
rectangle,  as  shown  in  the  drawing.  If  a square  be  added  to  this  rectangle,  giving 
the  familiar  shape  1.4472,  exactly  the  right  amount  of  space  is  obtained  to  restore 
a foot  of  the  normal  type.  The  body,  with  the  horizontal  handles,  is  enclosed  in 
a square.  The  body  without  the  handles  is  contained  in  the  rectangle  1.191 
( = .8396),  another  shape  which  occurs  frequently.  The  vase  omitting  the  three 
handles  is  placed  in  the  rectangle  1.528.  The  diameter  of  the  missing  foot  cannot 
of  course  be  determined  with  certainty.  In  the  drawings  it  has  been  assumed  to 
be  .4472.  This  is,  however,  probably  too  great.  The  relation  between  the  1.4472 
rectangle  and  the  1.528  rectangle  has  not  been  shown  in  the  geometrical  analysis. 

The  ratios  are : 


Height,  including  vertical  handle 

1.4472 

Diameter  of  lip 

. .5528 

Height,  omitting  vertical  handle  . 

1.2764 

Diameter  of  neck  at  bottom . . . 

. .3292 

Height  to  shoulder 

1.000 

Greatest  diameter  of  body 

. . .8396 

Height  of  neck 

.2236 

Diameter  of  bottom  of  body . . . 

, . .2236 

Width 

1.000 

Diameter  of  foot,  restored 

, . (.4472?) 

HYDRIA-KALPIS 


[ 109  ] 


GEOMETRY  OF  GREEK  VASES 


64  Kalpis.  Inv.  90.156.  Beazley,  V.  A.,  p.  148,  no.  VIII,  32.  Hambidge, 
p.  87.  The  death  of  Orpheus.  “Niobid  Painter.” 

Height,  0.403  m.  Width,  0.3785  m.  Diameter  of  body,  0.316  m.  Height  to 
neck,  0.3175  m. 

The  ratio  is  1.0652,  or  .618 + .4472,  showing  that  the  enclosing  area  is  com- 
posed of  a whirling  square  rectangle  plus  a \/5  rectangle.  In  the  drawing  the  v/5 
rectangle  has  been  placed  above  the  whirling  square  rectangle.  The  shoulder, 
marked  by  a painted  band,  is  at  half  the  height  of  the  \/5  rectangle,  and  the 
smallest  diameter  of  the  neck  equals  the  height  of  the  neck  and  lip.  The  vase 
up  to  the  shoulder,  and  omitting  the  handles,  is  contained  in  a square  (error: 
0.0015  m.).  The  top  of  the  maeander  band  on  which  the  figures  rest  is  at  half  the 
height  of  this  square.  The  ratios  are : 


Height 

. 1.0652 

Diameter  of  lip 

.382+ 

Height  of  neck  and  lip 

. .2236 

Smallest  diameter  of  neck 

.2236 

Height  to  shoulder 

. .8416 

Diameter  of  body 

.8416- 

Height  of  top  of  maeander  band  . 

. .4208 

Diameter  of  bottom  of  body . . . 

.236 

Width 

. 1.000 

Diameter  of  foot 

.382 

[ no  ] 


HYDRIA-KALPIS 


65  Inv.  03.792.  Kalpis,  Beazley,  V.  A.,  p.  162.  Danae  with  Perseus  seated 
in  chest;  a man  and  two  women  looking  at  them. 

Height,  0.404  m.  Width,  0.374  m.  Greatest  diameter  of  body,  0.308  m.  Diam- 
eter of  lip,  0.1585  m.  Diameter  of  foot,  0.1505  m. 

The  enclosing  rectangle  apparently  has  the  ratio  1.0787  ( = .927),  but  no  con- 
vincing analysis  has  been  found. 


[ m ] 


GEOMETRY  OF  GREEK  VASES 


66  Kalpis.  Inv.  10.183.  Eros  flying  towards  two  seated  women,  one  of  whom 
is  playing  with  a panther.  Free  style. 

Height,  0.1775  m.  Width,  0.1645  m.  Diameter  of  bowl,  0.133  m.  Diameter  of 
lip,  0.077  m.  Diameter  of  foot,  0.0745  m. 

The  enclosing  rectangle  has  the  ratio  .927,  or  1.0787.  The  proportions  are  as 
follows : 


Height 1.000 

Height,  omitting  foot. . . .927 

Width 927 

Diameter  of  bowl 750  = f 


Diameter  of  lip 427  = .927  — .500 

Smallest  diameter  of  neck . . .250  = £ 

Diameter  of  bottom  of  body  .309  = 927  -f-  3 
Diameter  of  foot 427  — = .927  — .500 


67  Kalpis.  Inv.  91.224.  Beazley,  V.  A.,  p.  175.  Hambidge,  p.  72,  fig.  14. 
Youth  between  woman  and  man.  Free  style. 

Height,  0.2775  m.  Width,  0.2485  m.  Diameter  of  bowl,  0.205  m. 

[ H2  ] 


HYDRIA-KALPIS 


If  the  width  is  regarded  as  unity,  the  height  has  the  ratio  1.118.  The  enclosing 
area  is  thus  composed  of  two  \/5  rectangles.  Without  its  handles  the  vase  fits 
inaccurately  the  familiar  area  1.382.  The  ratios  of  the  details  are  as  follows: 


Height 1.118 

Height  to  top  of  upper  painted  band . . . .9472 

Height  to  bottom  of  upper  painted  band  .736 

Height  to  top  of  lower  painted  band 4472 

Height  to  bottom  of  lower  painted  band  .382 


Width 1.000 

Diameter  of  bowl 809 

Diameter  of  Up 4472 

Diameter  of  bottom  of  body . .236 

Diameter  of  foot 382 


Another  kalpis,  Inv.  91.225,  Beazley,  V.  A.,  p.  175,  painted  by  the  same  hand, 
is  of  approximately  the  same  proportions,  but  none  of  the  details  fits  the  same 
scheme,  nor  has  it  been  possible  to  analyze  this  vase  in  terms  of  any  other  rec- 
tangle. Its  height  is  0.255  m.,  its  width  0.23  m. 


[ 113  ] 


GEOMETRY  OF  GREEK  VASES 


68  Kalpis.  Inv.  08.417.  Beazley,  V.  A.,  p.  121.  Hambidge,  p.  71.  The  death 
of  Argos.  Ripe  archaic  period. 

Height,  0.3685  m.  Width,  0.332  m.  Diameter  of  bowl,  0.28  m. 

The  enclosing  rectangle  has  the  ratio  1.118,  or  but  some  of  the  propor- 
tions of  details  are  more  simply  derived  from  the  rectangle  containing  the  vase 
without  its  handles.  This  has  the  ratio  1.309.  The  vase  up  to  the  neck  is  contained 
approximately  in  a square;  the  neck  and  lip  have  the  height  .309.  This  is,  how- 
ever, uncertain,  as  the  shoulder  is  not  marked  by  a painted  band,  as  often  happens. 
The  diameter  of  the  lip  equals  half  the  diameter  of  the  bowl.  The  relation  of  the 
over -all  rectangle  to  the  rectangle  of  the  bowl  is  indicated,  as  Mr.  Hambidge  has 
pointed  out,  by  the  diagonal  of  the  whole. 

The  chief  ratios,  regarding  the  total  width  as  unity,  are: 


Height 

1.118 

Diameter  of  lip 

..  .427 

Width 

1.000 

Diameter  of  bottom  of  body . . . . 

..  .2764 

Diameter  of  bowl 

854 

Diameter  of  foot 

..  .4045 

C 

114  ] 

HYDRIA-KALPIS 


69  Kalpis.  Inv.  95.22.  Dionysos,  Ariadne,  Eros,  Hermes,  a nymph  and  a 
seilen.  Free  style. 

Average  height,  0.315  m.  Width,  0.226  m.  to  0.233  m.  Diameter  of  bowl, 
0.192  m.  to  0.1935  m. 

In  the  drawing  the  vase  has  been  placed  in  a 1.382  rectangle.  The  average 
height  multiplied  by  .7236  (the  reciprocal  of  1.382)  gives  .227934.  The  height 
multiplied  by  .618  gives  .19467,  showing  that  the  vase  without  the  handles  is 
contained  fairly  accurately  in  a whirling  square  rectangle. 

The  ratios  are: 


Height 

1.382 

Smallest  diameter  of  neck.  . 

.250  (or  .236) 

Width 

1.000 

Diameter  of  bottom  of  body 

.236 

Diameter  of  bowl 

854 

Diameter  of  foot 

.382+ 

Diameter  of  lip 

500 

[ 115  ] 


GEOMETRY  OF  GREEK  VASES 


DEINOS  — KRATER 

Under  this  heading  are  included  vases  which  were  used  for  mixing  wine  and 
water.  The  collection  contains  three  examples  of  the  deinos,  or  lebes,  a bowl  with- 
out foot  or  handles  which  was  set  on  a tall  stand.  No.  70  is  an  example  of  Ionic 
pottery  of  the  sixth  century,  b.c.  ; no.  71,  which  has  the  same  main  proportions, 
belongs  to  the  later  Attic  red-figured  period.  No.  72,  of  the  black-figured  period, 
has  its  stand  preserved. 

The  large  krater  with  volute  handles,  from  Orvieto,  no.  73,  is  the  only  black- 
figured  krater  in  the  collection  which  is  worthy  of  study.  The  red-figured  kraters 
are  of  four  types.  No.  74  is  an  attic  example  of  the  column -handled  krater  which 
was  most  popular  in  Corinthian  times.  No.  75  is  included  as  an  example  of  the 
volute-handled  krater  as  it  appears  in  the  red-figured  period.  Its  scheme  of  pro- 
portions remain  obscure.  Three  kalyx  kraters  (nos.  76-78)  and  five  bell  kraters 
(nos.  79-83)  complete  the  list.  Nos.  81-83  have  not  been  analysed;  and  the 
analysis  of  no.  78  is  not  entirely  satisfactory. 

Nos.  72,  73,  76,  79  are  especially  worthy  of  study. 

70  Deinos.  Inv.  13.205.  Fairbanks,  American  Journal  of  Archaeology, 
XXIII,  1919,  p.  279.  Hambidge,  p.  127. 

Height,  0.2215  m.  Diameter,  0.277  m. 

The  ratio  is  1.250,  or  4:5.  The  enclosing  area  divides  into  twenty  squares,  the 
height  being  four  squares,  the  diameter  five  squares.  The  diameter  of  the  lip  is 
determined  by  the  intersection  of  the  diagonal  of  one  of  these  squares  with  the 
diagonal  of  two  and  a half  of  the  squares.  The  contour  of  the  bowl  has  an  interest- 
ing relation  to  the  small  squares,  as  is  shown  by  the  diagonals  of  the  two  pairs  of 
squares  at  the  lower  angle,  and  the  diagonal  of  half  the  square  at  the  upper  angle 
of  the  over-all  rectangle.  It  is  also  noteworthy  that  the  diameter  of  the  smallest 
of  the  painted  rings  equals  the  side  of  one  of  the  small  squares. 

71  Deinos.  Inv.  96.720.  Beazley,  V.  A.,  p.  175.  Athletic  scenes.  Freestyle. 

Height,  0.221  m.  Diameter,  0.276  m.  Diameter  of  lip,  0.221  m. 

This  deinos  of  the  late  red-figured  period  has  not  only  the  same  proportions  as 
the  preceding  example,  but  is  of  the  same  size,  the  difference  being  less  than  a 
millimetre.  The  diameter  of  the  lip  is  in  this  case  exactly  equal  to  the  height. 
The  variation  in  the  contour  is  greatest  at  the  shoulder  and  the  lower  part  of 
the  bowl. 


[ H6  ] 


DEIN  OS-KRATER 


[ ] 


GEOMETRY  OF  GREEK  VASES 


72  Deinos  and  Stand.  Inv.  90.154.  Ivy  pattern  on  the  neck  of  the  deinos  and 
on  the  upper  member  of  the  stand.  Inside  the  rim  of  the  deinos,  four  ships, 
painted  in  the  black-figured  technique.  Hambidge,  p.  81,  fig.  9;  photograph  opp. 

p.  82. 

Height  of  deinos  and  stand  together,  approximately  0.5585  m.  Height  of 
deinos,  0.3165  m.  Height  of  stand,  0.299  m.  Diameter  of  lip  of  deinos,  0.281  m. 
Largest  diameter  of  deinos,  0.388  m.  Diameter  of  top  of  stand,  0.276  m.  Diam- 
eter of  foot,  0.311  m. 

In  view  of  the  fact  that  two  separate  pieces  of  pottery  are  combined  to  form 
one  whole,  and  that  the  stand  is  not  exactly  perpendicular,  some  of  these  measure- 
ments are  uncertain.  The  two  pieces  together,  however,  fit  a simple  scheme  fairly 
accurately.  The  containing  shape  is  made  up  of  a square  and  a \/5  rectangle 
(1.4472).  The  deinos  alone  is  placed  in  a rectangle  which  has  nearly  the  ratio 
.809,  i.  e.,  it  is  composed  of  two  whirling  square  rectangles  placed  vertically  side  by 
side.  The  diameter  of  the  lip  (.7236)  and  the  diameter  of  the  bowl  where  it  joins 
the  lip  (.618)  are  simply  derived  from  the  .809  rectangle.  The  former  also  equals 
half  of  the  total  height  (.7236X2  = 1.4472).  The  portion  of  the  over-all  rec- 
tangle in  which  the  stand  is  contained  has  the  ratio  .7764,  or  .500+. 2764.  The 
ratio  .500  is  represented  by  two  squares.  The  reciprocal  of  .2764  is  3.618,  or  two 
squares  plus  a horizontal  whirling  square  rectangle.  The  height  of  the  upper 
member  of  the  stand  (two  toruses  with  a scotia  between)  is  half  of  .2764,  or  .1382. 
Many  of  the  proportions  of  the  details  of  the  stand  are  simply  expressed  in  terms 
of  a whirling  square  rectangle  applied  at  the  bottom  of  the  over-all  shape.  The 
smallest  diameter  of  the  stand  is  close  to  .236.  The  diameter  of  the  top  of  the 
stand  is  .236X3,  or  .708.  This  is  also  the  diameter  of  the  foot  above  the  torus  at 
the  bottom.  The  height  of  the  foot  is  half  of  .236,  or  .118.  The  height  of  the  foot 
and  stem  (up  to  the  first  moulding)  is  .4472.  The  greatest  diameter  of  the  foot  is 
equal  to  the  height  of  the  deinos  (.809).  As  Mr.  Hambidge  has  noted,  a rectangle 
with  the  diameter  of  the  foot  as  the  short  side  and  the  total  height  as  the  long  side 
has  the  ratio  1.7888,  or  .4472X4. 


[ 118  ] 


DEIN  OS-KRATER 


[ H9] 


GEOMETRY  OF  GREEK  VASES 


KRATER 

BLACK-FIGURED  PERIOD 

73  Volute-handled  Krater.  Inv.  90.153.  Ann.  Rep.  1890,  p.  16,  no.  1.  Ham- 
bidge,  p.  73. 

Average  height  of  handles,  0.773  m.  Width  (distance  apart  of  handles) 
0.684  m.  Height  without  handles,  0.6843  m.  Height  to  shoulder,  0.494  m. 
Diameter  of  lip,  0.559  m.  Diameter  of  shoulder,  0.428  m.  Diameter  of  bowl, 
0.540  m.  Diameter  of  bottom  of  bowl  0.181  m.  Diameter  of  foot,  0.307  m. 

The  greatest  height  can  be  determined  only  approximately,  since  the  four 
volutes  of  the  handles  are  all  at  different  levels.  The  ratio  1.118  gives  a height  of 
0.7644  m.,  or  about  8 mm.  less  than  the  average  height  of  the  handles.  The  pro- 
portions of  several  details  (diameter  of  lip,  division  in  neck,  shoulder,  step  on  foot, 
height  of  foot)  are  obtained  simply  from  divisions  of  this  1.118  rectangle.  Other 
details  can  be  more  directly  derived  from  the  square  in  which  the  krater  is  en- 
closed if  the  projection  of  the  handles  above  the  lip  is  omitted.  This  square  is 
subdivided  in  the  drawing  by  placing  \/ 5 rectangles  vertically  and  horizontally 
across  its  centre.  The  resulting  figures  are  clearly  shown  in  the  second  of  the  small 
diagrams. 

It  is  noteworthy  that  the  krater  without  its  handles  is  contained  in  the  familiar 
rectangle  1.236  (=  .809),  i.  e.,  two  whirling  square  rectangles  placed  horizontally, 
one  above  the  other.  The  body,  considered  separately,  i.  e.,  omitting  the  neck, 
lip,  and  foot,  is  contained  in  a similar  area,  the  two  whirling  square  rectangles  in 
this  case  being  placed  vertically,  side  by  side.  Cf.  the  third  and  fourth  of  the  small 
diagrams. 

The  study  of  this  krater  is  interesting  because  of  its  monumental  size  and 
because  of  the  accuracy  with  which  most  of  the  details  are  expressible  in  simple 


ratios,  as  follows: 

Height  of  handles  (inaccurate) 1.118 

Height  of  lip 1.000 

Height  to  lower  line  of  lip 9655 

Height  to  division  in  neck  (A)  upper  line 868 

( B ) lower  line 8618 

Height  to  shoulder 7236 

Height  to  painted  band  near  bottom  of  body 191 

Height  of  foot  (A)  upper  line 090 

( B ) lower  fine 0854 

Width  (handles) 1.000 

Diameter  of  lip  (inaccurate) 809 

Diameter  of  division  in  neck 691 

Diameter  of  shoulder  (inaccurate) 618 

Greatest  diameter  of  bowl 7888 

Diameter  of  bottom  of  bowl . 264 

Diameter  of  step  on  foot 382 

Diameter  of  foot 4472 


[ 120  ] 


KRATER 


[ 121  3 


GEOMETRY  OF  GREEK  VASES 


RED-FIGURED  PERIOD 

74  Column  Krater.  Inv.  89.274.  Beazley,  V.  A.,  p.  134,  no.  I,  6.  (A)  Woman 
with  torch,  and  man.  (B)  Youth  between  women.  “ Orchard  Painter.” 

Height,  0.355  m.  Width,  0.365  m.  Diameter  of  bowl  0.282  m. 

The  enclosing  area  is  very  close  to  the  rectangle  1.0225,  or  .978,  made  up  of  a 
vertical  whirling  square  rectangle  with  four  horizontal  whirling  square  rectangles 
beside  it.  In  the  drawing  most  of  the  proportions  are  obtained  from  subdivisions 
of  the  large  whirling  square  rectangle.  The  diameter  of  the  foot  is  contained  in 
the  central  square  of  a \/5  rectangle  described  on  the  base  of  the  over-all  shape. 

The  ratios  appear  in  a more  familiar  form  when  the  width  is  regarded  as  unity: 


Height 

978 

Diameter  at  shoulder 

..  .528 

Height  to  shoulder 

691 

Largest  diameter  of  bowl 

, ..  .764 

Width 

1.000 

Diameter  of  bottom  of  bowl . . . . 

...  .309 

Diameter  of  lip 

820 

Diameter  of  foot 

. . .4472 

[ 122  ] 


KRATER 


75  Volute-handled  Krater.  Inv.  00.347.  Beazley,  V.  A.,  p.  151.  (A)  Apollo, 
Artemis,  and  Leto.  ( B ) Three  women  sacrificing  at  an  altar.  “ Painter  of  the 
Berlin  Nike  Hydria.” 

Greatest  height,  0.525  m.  Width,  0.3915  m.  Height  of  lip,  0.4482  m.  Diame- 
ter of  lip  and  of  bowl,  0.322  m.  Height  of  shoulder,  0.343  m.  Diameter  of  foot, 
0.175  m. 

No  plausible  analysis  of  this  krater  has  been  found. 


[ 123  ] 


GEOMETRY  OF  GREEK  VASES 


76 


76  Kalyx  Krater.  Inv.  95.23.  (A)  Zeus  pursuing  Aegina;  second  woman  flee- 
ing. ( B ) An  old  man  standing,  with  sceptre,  is  approached  by  a woman  with  hands 
extended.  Between  them  an  altar.  Ripe  archaic  period. 

Height,  0.438  m.  Diameter,  0.4785  m.  Diameter  of  foot,  0.2195  m. 

If  the  height  is  taken  as  0.439  m.,  which  equals  twice  the  diameter  of  the  foot, 
the  containing  rectangle  has  exactly  the  ratio  1.0902.  This  is  made  up  of  the 
ratios  618  and  .4722.  The  latter  is  usually  written  .472  in  this  book. 

The  ratios  are  as  follows : 


Height 

1.000 

Diameter  of  bottom  of  lip 

..  .8992 

Height  to  lip 

.882 

Diameter  of  lower  member 

..  .708 

Height  of  lip 

.118 

Diameter  of  bottom  of  body . . . . 

. . .2764 

Height  of  lower  member  of  krater 
Diameter 

.4045 

1.0902 

Diameter  of  foot 

..  .500 

The  krater  without  the  lip  is  contained  in  two  whirling  square  rectangles  as  is 
shown  by  the  diagonals.  The  intersections  of  these  diagonals  with  the  top  of  the 
lower  member  fix  the  diameter  of  the  foot. 


[ 124  ] 


KRATER 


77  Kalyx  Krater.  Inv.  03.817.  Beazley,  V.  A 
Aegina.  (B)  Draped  youth.  “Achilles  Painter.” 

Height,  0.2405  m.  Diameter, 

0.244  m. 

In  the  drawing  the  krater  has 
been  placed  in  a square,  the 
height  and  diameter  being  taken 
as  0.242  m.  If  \/2  rectangles  are 
applied  at  each  side,  the  foot  is 
contained  in  the  overlapping 
portion  (.4142).  The  diameter  of 
the  bottom  of  the  body  is  half 
that  of  the  foot  (.2071).  The 
height  of  the  lower  member  of 
the  krater  is  slightly  less  than 
.4142.  Other  details  are  ac- 
counted for  by  geometrical  divi- 
sions of  the  \/2  rectangle.  But 
the  analysis  is  not  altogether 
convincing. 


p.  163.  (A)  Zeus  pursuing 


78  Kalyx  Krater. 

( B ) Draped  youth. 

Height,  0.189  m.  Diameter, 
0.194  m. 

The  ratio  is  assumed  to  be 
1.0225  (.978),  the  height  being 
regarded  as  unity.  The  diameter 
of  the  foot  is  obtained  by  apply- 
ing whirling  square  rectangles  at 
each  side,  the  foot  being  con- 
tained in  the  overlapping  portion 
(.4045).  The  diameter  of  the 
bottom  of  the  body  is  .20225, 
or  half  of  .4045.  The  height  of 
the  lower  member  is  also  .4045. 
No  other  simple  ratios  have  been 
found,  and  the  analysis  is  not  to 
be  regarded  as  satisfactory. 


Inv.  03.796.  (A)  Youth  with  a goat,  approaching  Hermes. 


[ 125  ] 


GEOMETRY  OF  GREEK  VASES 


79  Bell  Krater.  Inv.  10.185.  Beazley,  V.  A.,  p.  113.  Hambidge,  p.  88.  ( A ) 
Artemis  killing  Aktaeon.  (B)  Pan  pursuing  a shepherd.  “ Pan  Painter.” 

The  height  varies  between  0.370  m.  and  0.372  m.,  the  diameter  between 
0.420  m.  and  0.428  m.  The  diameter  of  the  foot  is  0.211  m.  If  the  diameter  of  the 
vase  be  taken  as  twice  the  diameter  of  the  foot,  0.422  m.,  and  the  height  be  taken 
as  0.372  m.,  the  over -all  rectangle  has  the  ratio  1.1338,  or  .882.  The  latter  is 
.382+. 500,  a simple  shape  which  accounts  for  all  the  details. 

The  main  proportions  are  made  clear  in  the  four  small  diagrams.  In  diagram  A 
the  two  squares  (.500)  are  placed  above  the  .382  shape  which  may  be  expressed  as 
a square  flanked  by  two  whirling  square  rectangles  on  each  side.  The  diameter  of 


the  bottom  of  the  bowl  equals  the  side  of  this  square,  or  .382.  In  diagram  B the 
two  squares  are  placed  below;  perpendiculars  dropped  from  their  centers  deter- 
mine the  diameter  of  the  foot.  The  square  .382  is  used  above  as  the  central  square 
of  a rectangle  the  ends  of  which  determine  the  largest  diameter  of  the  bowl. 
Diagram  C shows  that  diagonals  of  half  the  shape  intersect  the  sides  of  the  rec- 
tangle enclosing  the  bowl  at  the  level  of  its  top.  This  proves  that  the  vase  without 
the  lip  and  handles  is  of  the  same  shape  as  the  complete  vase,  a phenomenon  which 
appears  also  in  the  kantharos,  no.  121.  Diagram  D shows  that  the  height  of  the 
krater  to  the  level  of  the  greatest  projection  of  the  lip  is  equal  to  the  diameter  of 


[ 126  ] 


KRATER 


79 


the  bowl.  The  height  of  the  foot,  and  the  levels  of  the  top  and  bottom  of  the 
maeander  band  on  which  the  figures  on  side  B rest  are  obtained  by  simple  sub- 
divisions of  the  square  .382,  as  is  shown  in  the  large  drawing. 

The  ratios,  regarding  the  diameter  as  unity,  are  as  follows : 


Height 

.882 

Height  of  foot 

. . .090 

Height  to  greatest  projection  of  lip 

.854 

Diameter 

...  1.000 

Height  without  lip 

.7532 

Diameter  of  top  of  bowl 

. . .854 

Height  to  top  of  maeander  band. . 

.236 

Diameter  of  bottom  of  bowl . . . . 

. . .382 

Height  to  bottom  of  maeander  band 

.191 

Diameter  of  foot 

. . .500 

C 127  ] 


GEOMETRY  OF  GREEK  VASES 


80 


80  Bell  Krater.  Inv.  00.348.  (A)  Athena  playing  the  flutes  in  the  centre  of  a 
group.  ( B ) Three  seilens  and  a maenad.  Free  style. 

Height,  0.34  m.  Diameter,  0.387  m. 

The  rectangle  has  the  ratio  .882  (1.1338),  i.  e.,  5.000+.382.  The  proportions 


are: 

Height 882 

Height  to  top  of  bowl 809 

Height  of  foot 090  — 

Diameter 1.000 

Diameter  at  top  of  bowl 764  = .382  X 2 

Diameter  of  bowl  at  level  of  handles 708 

Diameter  of  bottom  of  bowl 382 

Diameter  of  foot 416 


[ 128  ] 


KRATER 


81  Bell  Krater.  Inv.  95.24.  Beazley,  V.  A.,  p.  184.  Sacrifice  of  a sheep.  Free 
style. 

Height,  0.4136  m.  Width  (handle  to  handle),  0.423  m.  Diameter  of  lip, 
0.411  m. 

The  containing  shape  appears  to  be  a 1.0225  rectangle.  If  the  diameter  of  the 
lip  is  regarded  as  the  width,  the  shape  is  very  close  to  a square.  No  satisfactory 
analysis  of  the  details  has  been  found. 


[ 129  ] 


GEOMETRY  OF  GREEK  VASES 


82  BellKrater.  Inv.  00.346.  Beazley,  V.  A.,  p.  173.  (A)  Aktaeon  attacked 
by  dogs  in  the  presence  of  Artemis,  Lyssa,  and  Zeus.  (B)  Youth  standing  between 

two  women.  “Lykaon 
Painter.” 

Height,  0.377  m. 
Width  (handles) , 0.441 
m.  Diameter  of  lip, 
0.413  m.  Diameter  of 
foot,  0.187  m. 

The  over-all  ratio  is 
1.1708  (=  .854).  The 
diameter  of  the  foot  is 
close  to  half  the  height. 
No  satisfactory  anal- 
ysis has  been  found. 


82 


83  BellKrater.  Inv. 
76.50.  (A)  Bearded 
man,  woman  with 
flutes  and  youth.  ( B ) 
Three  youths.  Free 
style. 

Height,  0.2935  m. 
Width  (handles),  0.341 
m.  Diameter  of  lip, 
0.322  m. 

The  enclosing  rec- 
tangle has  the  ratio 
1.1618,  which  is  that 
of  the  ground  plan  of 
the  Erechtheum.  No 
significant  proportions 
have  been  noted. 


[ 130  ] 


PSYKTER 


RED-FIGURED  PSYKTER  (WINE-COOLER) 

84  Inv.  01.8019.  Beazley,  V.  A.,  p.  29,  no.  V,  8.  Hambidge,  p.  99.  Athletes 
and  trainers. 

Height,  0.3435  m.  Diameter,  0.269  m. 

The  ratio  is  1.2764,  expressed  geometrically  as  a square  with  a 3.618  (.2764) 
rectangle  added. 

The  proportions  of  the  lower  part  of  the  vase  can  be  expressed  in  simple  terms 
of  a whirling  square  rectangle  applied  at  the  bottom  of  the  containing  area.  The 


top  of  this  rectangle  coincides  with  the  painted  band  on  which  the  figures  stand. 
The  diameter  of  the  lip  equals  one-half  of  the  diameter  of  the  vase.  The  height 
and  diameter  of  the  shoulder  are  clearly  expressed  in  terms  of  a .309  rectangle 
applied  at  the  top  of  the  containing  area. 


C 131  ] 


GEOMETRY  OF  GREEK  VASES 


The  ratios  are : 

Height 1.2764 

Height  to  shoulder 1.0854 

Height  of  neck  and  lip 191 

Height  to  painted  band  below  figures  .618 
Diameter 1.000 


Diameter  of  lip 500 

Diameter  of  shoulder 382 

Smallest  diameter  of  stem 382 

Diameter  of  ring  at  top  of  foot . . .4472 

Diameter  of  foot 5669 


OINOCHOE  ( WINE  JUG) 

Under  this  name  are  included  sixteen  vases,  four  of  the  black-figured  and 
twelve  of  the  red-figured  period.  Of  the  black-figured  examples  three  have  a tre- 
foil lip;  in  two  of  them  the  line  of  the  shoulder  is  accentuated  plastically;  in  the 
third  it  is  marked  only  by  a painted  band.  The  fourth  is  of  the  shape  to  which  the 
name  “ olpe  ” is  often  given.  It  has  a large  cylindrical  mouth,  and  the  neck 
merges  gradually  into  the  body.  The  red-figured  examples  include  three  oinochoai 
with  trefoil  lip,  but  differing  widely  from  one  another  in  proportions,  and  nine 
vases,  which  might  be  designated  by  the  name  "olpe,”  in  that  all  of  them  have 
large  cylindrical  mouths,  but  which  are  otherwise  very  dissimilar.  The  last  three 
examples,  because  of  their  shape  and  small  size,  appear  to  have  been  drinking 
mugs  rather  than  wine  pitchers. 


BLACK-FIGURED  PERIOD 

85  Oinochoe  with  Trefoil  Lip.  Inv.  98.924.  Herakles  leading  a sphinx. 

Height  of  handle,  0.173  m.  Height  of  lip,  0.157  m.  Diameter,  0.107  m. 

The  vase  is  enclosed  in  a whirling  square  rectangle,  but  the  proportions  of  de- 
tails are  obtained  more  clearly  from  the  rectangle  1.472,  which  is  obtained  by 
dividing  the  diameter  into  the  height  of  the  lip.  If  a horizontal  whirling  square 
rectangle  is  applied  at  the  bottom  of  the  over-all  shape,  the  intersection  of  its 
diagonal  with  the  diagonal  of  the  1.472  rectangle  fixes  the  diameter  of  the  bottom 
of  the  body.  The  diameter  of  the  foot  is  obtained  from  the  intersection  of  the 
1.472  diagonal  with  the  horizontal  line  at  half  the  height  of  the  whirling  square 
rectangle.  The  diameter  of  the  shoulder  is  given  by  the  intersection  of  the  1.472 
diagonal  with  the  horizontal  line  at  the  level  of  the  shoulder.  This  level  is  at  the 
height  1.090,  or  1.472  — .382.  The  height  of  the  neck  and  lip  is  thus  .382,  and  the 
width  of  the  lip  is  seen  to  be  .382  X 2,  or  .764.  The  smallest  diameter  of  the 
neck  is  also  .382;  and  the  width  of  the  spout  is  .236,  i.  e.,  it  is  enclosed  in  the 
whirling  square  rectangle  in  the  centre  of  the  .382  (2.618)  shape.  The  ratios  are: 


Height  to  top  of  handle 1.618 

Height  to  top  of  lip 1.472 

Height  of  neck  and  lip  alone 382 

Height  to  shoulder 1.090 

Diameter 1.000 

Width  of  lip 764 


Width  of  spout 236 

Smallest  diameter  of  neck 382 

Diameter  of  shoulder 481 

Diameter  of  bottom  of  body 4086 

Diameter  of  foot 5801 


[ 132  ] 


OINOCHOE 


C 133  ] 


GEOMETRY  OF  GREEK  VASES 


86  Oinochoe  with  Trefoil  Lip.  Inv.  99.527.  Butcher  cutting  up  meat. 
Height,  with  handle,  0.25  m.;  without  handle,  0.211  m.  Diameter,  0.154  m. 
This  vase  is  also  enclosed  in  a whirling  square  rectangle,  and  its  analysis  is  remark- 
ably simple. 

The  proportions  can  be  expressed  arithmetically  as  follows : 


Height  to  top  of  handle 

. 1.618 

Width  of  spout 

. .236 

Projection  of  handle  above  lip  . 

. .250 

Width  of  handle 

. .191 

Height  to  top  of  lip 

. 1.368 

Diameter  of  shoulder 

. .500 

Height  to  shoulder 

. .9635 

Diameter  of  bottom  of  body  . . . 

. .4472 

Diameter 

Width  of  mouth 

. 1.000 
. .750 

Diameter  of  foot 

. .5669 

[ 134  ] 


OINOCHOE 


87  Oinochoe  with  Trefoil  Lip.  Inv.  13.74.  Dionysos  reclining  under  a grape- 
vine. Height  to  top  of  handle,  0.228  m.;  to  top  of  lip,  0.216  m.  Diameter, 
0.1335  m. 

The  over-all  ratio  is  1.708.  If  the  projection  of  the  handle  above  the  lip  is  dis- 
regarded the  ratio  is  1.618.  This  whirling  square  rectangle  has  been  used  in  the 
analysis  because  of  its  greater  simplicity.  The  ratios  are : 


Height  to  top  of  handle 

....  1.708 

Width  of  lip 

..  .764 

Height  to  top  of  lip 

....  1.618 

Smallest  diameter  of  neck 

..  .382 

Height  to  shoulder 

. . . . 1.146 

Diameter  of  shoulder 

..  .528 

Height  of  neck  and  lip 

472 

Diameter  of  bottom  of  body . . . . 

..  .333 

Diameter 

....  1.000 

Diameter  of  foot 

..  .5669 

[ 135  ] 


GEOMETRY  OF  GREEK  VASES 


88  Olpe.  Inv.  03.783.  Herakles  in  the  beaker  of  Helios. 

Height  to  top  of  handle,  0.219  m. ; to  top  of  lip,  0.2096  m.  Diameter,  0.113  m. 
If  the  handle  is  included  the  enclosing  rectangle  is  of  the  shape  1.927,  = 1.000+ 
.618+. 309.  Without  the  handle  the  ratio  is  1.854,  = .618X3. 

The  proportions  are  as  follows : 

Height  to  top  of  handle 1.927  Smallest  diameter  of  neck 618 

Height  to  top  of  lip 1.854  Diameter  of  bottom  of  body 4472 

Diameter 1.000  Diameter  of  foot 6584 

Diameter  of  lip 7888 


[ 136  ] 


OINOCHOE 


RED-FIGURED  PERIOD 


89  Oinochoe  with  Trefoil  Lip.  Inv. 
97.370.  Beazley,  V.  A.,  p.  145,  no.  24. 
Apollo  and  Artemis.  “Altamura 
Painter.” 

Height  to  top  of  handle,  0.3465  m. 
Diameter,  0.168  m. 

The  rectangle  has  the  ratio  2.0652, 
or  1.000 + -618 + .4472.  The  propor- 
tions are : 


Height 

Height  to  lip  at  ends  . 
Height  to  shoulder  . . 

Diameter 

Diameter  of  Up 

Distance  apart  of  ends 

of  handle 

Width  of  handle 

Width  of  spout 

Smallest  diameter  of 

neck 

Diameter  of  shoulder . 
Diameter  of  bottom 

of  body 

Diameter  of  foot .... 


2.0652 

1.7562  = 2.0652 -.309 
1.2764 
1.000 
.666 

.618 

.1545  = .309-4-2 
.309 


.382- 

.5528 

.427 

.618 


[ 137  2 


GEOMETRY  OF  GREEK  VASES 


90  Oinochoe  with  Trefoil  Lip.  Inv.  03.794.  Hambidge,  p.  56.  Young  warrior 
standing  before  seated  old  man. 

Height  to  top  of  handle,  0.186  m.  Diameter,  0.1325  m. 

The  vase  is  contained  in  a y/2  rectangle.  The  ratios  are: 


Height  to  top  of  handle 1.4142 

Height  to  top  of  spout 1.3685 

Height  to  shoulder 1 .0355 

Height  to  bottom  of  painted  band  on  shoulder 1.000 

Diameter 1.000 

Width  of  lip 6465  — 

Smallest  diameter  of  neck 333 

Diameter  of  shoulder 4142 

Diameter  of  bottom  of  body 60952 

Diameter  of  foot 6465  — 


[ 138  ] 


OINOCHOE 


91  Oinochoe  with  Trefoil  Lip  and  Squat  Body.  Inv.  00.352.  Beazley,  V.  A., 
p.  177.  Seilen  and  two  women.  Free  style. 

Height,  0.212  m.  Diameter,  0.1725  m. 

These  dimensions  give  the  ratio  1.236,  i.  e.,  618X2,  or  two  whirling  square 
rectangles. 

The  proportions  are : 

Height 1.236  Smallest  diameter  of  neck 5000 

Diameter 1.000  Diameter  at  bottom  of  body 708 

Width  of  lip 7236  Diameter  of  foot 7888  — 


[ 139  ] 


GEOMETRY  OF  GREEK  VASES 


92-95  Four  Oinochoai  with  Cylindrical  Mouth.  Inv.  13.192,  13.191,  13.196, 
13.197.  Beazley,  V.  A.,  p.  157.  92.  Two  komasts.  93.  Two  athletes.  94.  Greek 
and  Persian.  95.  Seilen  and  maenad.  Early  free  style.  The  paintings,  according 
to  Beazley,  are  perhaps  by  the  painter  of  the  Chicago  stamnos. 

These  four  jugs  should  be  studied  as  a group.  The  paintings  are  all  by  the 
same  hand,  and  the  vases  are  evidently  from  the  same  factory.  They  resemble  one 
another  very  closely  in  dimensions,  contour,  and  perfection  of  workmanship. 
Their  walls  are  so  thin  as  to  make  their  weight  astonishingly  slight.  All  are 
covered  by  a deep,  lustrous  black  varnish. 

The  actual  dimensions  are  given  in  table  1. 

The  principal  differences  are  seen  to  be  in  the  height  of  the  body  which  is  equal 
to  the  diameter  in  nos.  4 and  5,  less  by  4 mm.  in  no.  6,  and  by  9 mm.  in  no.  7.  The 
diameter  of  the  lip  remains  almost  the  same,  but  the  diameter  of  the  foot  varies 
considerably.  The  mouldings  of  the  lip  and  foot  are  nowhere  exactly  duplicated. 
Nos.  92-94  have  one  indented  ring  around  the  neck  below  the  lip;  no.  95  has  two 
such  rings.  All  four  vases  have  a band  with  a painted  tongue  pattern  marking  the 
shoulder,  and  a maeander  band  encircling  the  body,  as  a ground  for  the  pictures. 

[ 140  ] 


OINOCHOE 


TABLE  1.  DIMENSIONS 


92 

93 

94 

95 

Height  to  top  of  handle 

0.247  m. 

0.245  m. 

0.240  m. 

0.229  m. 

Height  to  top  of  lip 

0.200 

0.199 

0.193 

0.1855 

Height  to  top  of  painted  band  on  shoulder . . . 

0.1525 

0.152 

0.1475 

0.141 

Diameter 

0.1525 

0.152 

0.1515 

0.150 

Diameter  of  lip 

0.098 

0.097 

0.0965 

0.094 

Smallest  diameter  of  neck 

0.078 

0.078 

0.0765 

0.065 

Diameter  of  bottom  of  body 

0.057 

0.061 

0.0575 

0.0685 

Diameter  of  foot 

0.082 

0.086 

0.0825 

0.086 

In  view  of  the  fact  that  the  diameter  decreases  from  0.1525  m.  in  no.  92  to 
0.150  m.  in  no.  95,  the  correspondences  and  variations  in  proportions  can  be 
more  easily  noted  in  the  following  table  of  ratios,  in  which  the  diameter  is  taken 
as  unity: 


[ 141  ] 


GEOMETRY  OF  GREEK  VASES 


TABLE  2.  RATIOS 


92 

93 

94 

95 

Height  to  top  of  handle 

1.618 

1.618- 

1.5854 

1.528 

Height  to  top  of  lip 

1.309 

1.309 

1.2764 

1.236 

Height  to  top  of  painted  band  on  shoulder . . . 

1.000 

1.000 

.9674 

.944 

Height  to  indented  ring  on  neck 

1.2135 

1.2135 

1.1809 

1.146 

Projection  of  handle  above  lip 

.309 

.309- 

.309 

.292 

Height  of  neck  and  lip 

.309 

.309 

.309- 

.292 

Height  to  top  of  maeander  band  on  body .... 

.4045 

.4045 

.382 

Diameter 

1.000 

1.000 

1.000 

1.000 

Diameter  of  lip 

.618  + 

.618+ 

.618+ 

.618+ 

Smallest  diameter  of  neck 

.500+ 

.500+ 

.500+ 

.500 

Diameter  of  bottom  of  body 

.382 

.4045 

.382 

.4472 

Diameter  of  foot 

.528+ 

.5669 

.5528 

.5669 

[ 142  ] 


OINOCHOE 


The  table  shows  that  nos.  92  and  93  are  enclosed  in  a whirling  square  rec- 
tangle. And,  with  the  exception  of  the  foot,  the  proportions  of  details  fit  exactly 
the  same  simple  scheme  obtained  from  the  containing  shape.  The  body  in  each 
case  is  a perfect  square.  The  portion  above  the  shoulder  is  enclosed  in  the 
reciprocal  of  the  over-all  rectangle,  and  the  lip  is  at  half  the  height  of  the  recip- 
rocal. The  diameter  of  the  neck  is  half  the  diameter  of  the  body.  The  indented 
ring  on  the  neck  is  at  three-quarters  of  the  total  height,  the  top  of  the  maeander 
band  is  at  one-quarter  of  the  height.  No.  94  is  identical  with  nos.  92  and  93  as 
regards  its  handle,  neck  and  lip,  even  to  the  placing  of  the  indented  ring,  but  the 
height  of  the  body  has  been  reduced.  No.  95  is  of  a still  more  squat  shape,  and  the 
height  of  neck  and  handle  has  been  reduced  proportionally  to  that  of  the  body. 


[ 143  ] 


GEOMETRY  OF  GREEK  VASES 


96  Olpe.  Inv.  13.93.  Comic  actor. 

Height  to  top  of  lip,  0.182  m.  Diameter,  0.129  m. 

The  ratio  is  that  of  the  y/2  rectangle,  1.4142.  The  proportions  of  details  are: 


Height  to  top  of  lip 

1.4142 

Smallest  diameter  of  neck 

. . .5858 

Height  to  top  of  ring  on  shoulder. 

.9428 

Diameter  of  shoulder 

. . .6394 

Height  to  bottom  of  ring  on  shoulder 

.9142 

Diameter  of  bottom  of  body . . . 

. . .6394 

Diameter 

Diameter  of  lip 

1.000 

.7777 

Diameter  of  foot 

. . .6666 

OINOCHOE 


97  Olpe.  Inv.  76.51.  Hambidge,  p.  128.  No  painted  decoration. 

Height,  0.1995  m.  Diameter,  0.1235  m. 

These  figures  are  very  close  to  2.000  and  1.236,  which  give  the  ratio  of  the 
whirling  square  rectangle,  1.618.  The  ratios  are: 

Height  to  top  of  lip 1.618  Smallest  diameter  of  neck 528 

Diameter 1.000  Diameter  of  foot 708 

Diameter  of  lip 7888 


C 145  ] 


GEOMETRY  OF  GREEK  VASES 


98  Olpe.  Inv.  95.56.  Beazley,  V.  A.,  p.  158.  Dancing  seilen.  “Euaion 
Painter.” 

Height,  0.1065  m.  Diameter,  0.10  m. 

The  ratio  is  obviously  1.0652,  i.  e.,  .618+. 4472,  a whirling  square  rectangle 
plus  a \/5  rectangle.  Cf.  the  oinochoe  no.  89,  whose  ratio  is  2.0652. 

On  account  of  the  diminutive  size  the  geometrical  analysis  of  some  of  the  de- 
tails has  been  omitted.  The  ratios  are: 


Height 1.0652 

Height  omitting  base 1.000 

Height  of  lower  termination  of 

handle 618 

Diameter 1.000 


Diameter  of  lip 9472? 

Smallest  diameter  of  neck 764 

Diameter  of  bottom  of  body 8292? 

Diameter  of  foot 8944 

Width  of  handle 236 


[ 146  ] 


OINOCHOE 


99  Olpe.  Inv.  97.606.  Eros 
flying. 

Height  and  diameter,  0.075  m. 
The  vase  is  enclosed  in  a 
square. 


100  Olpe.  Inv.  00.339.  Beaz- 
ley,  V.  A.,  p.  92.  Youth  dancing 
with  castanets;  woman  playing 
flutes.  “ Brygos  Painter.” 

Height  to  top  of  lip,  0.0805  m. 
Diameter,  0.0905  m. 

The  ratio  is  1.118,  or  The 


proportions  are: 

Height 1.000 

Height  to  shoulder 750 

Diameter 1.118 

Diameter  of  lip  ...  . 1.118  — 

Diameter  of  neck 882 

Diameter  of  base 944 


100 


[ 147  ] 


GEOMETRY  OF  GREEK  VASES 


SKYPHOS 


This  form  of  drinking  cup  has  a long  history.  In  addition  to  the  fourteen  Attic 
examples  here  published,  three  earlier  skyphoi  are  illustrated  for  the  sake  of  com- 
parison. The  first  is  of  the  so-called  Proto-Corinthian  style,  — a small  cup  of 
very  fine  workmanship  with  walls  of  egg-shell  thinness,  and  decoration  consisting 
of  numerous  horizontal  bands.  The  second  is  a Corinthian  skyphos.  The  third 
example,  of  the  same  general  shape  as  the  preceding  one,  is  presumably  to  be 
classed  as  Corinthian. 


101  Inv.  03.809. 

Height,  0.0875  m.  Width, 
0.138  m.  Diameter  of  bowl, 
0.1005  m.  Diameter  of  foot, 
0.035  m.  The  diameter  of  the  bowl 
is  two  and  one-half  times  that 
of  the  foot  (0.035X2.5  = 0.875). 
The  total  width  is  close  to  four 
times  the  diameter  of  the  foot 
(0.035X4  = 0.140).  If  any  scheme 
of  proportion  was  used  it  appears 
to  have  been  of  the  “static” 
type. 


102  Inv.  95.14.  Fowler  and  Wheeler,  Greek  Archaeology , p.  449,  fig.  365. 
(. A ) Two  warriors.  ( B ) Two  leopards  with  a palmette-lotus  design  between  them. 

Height,  0.091  m.  Width,  0.198  m.  Diameter  of  bowl,  0.139  m.  Diameter  of 
foot,  0.07  m.  The  over-all  ratio  is  close  to  2.1708.  The  skyphos  without  the 
handles  fits  a 1.528  rectangle,  and  the  diameter  of  the  foot  is  slightly  greater  than 
one-half  the  diameter  of  the  bowl. 


[ 148  ] 


SKYPHOS 


103  Inv.  97.366.  (A)  Two  lions.  ( B ) Palmette-lotus  design,  fine  workman- 
ship. 

Height,  0.112  m.  Width,  0.2315  m.  Greatest  diameter  of  bowl,  0.165  m. 
Diameter  of  lip,  0.162  m.  Smallest  diameter  of  bowl,  0.793  m.  Diameter  of  foot, 
0.095  m. 

The  enclosing  area  apparently  has  the  ratio  2.0652  (cf.  the  oinochoe  no.  89). 
If  the  handles  are  omitted  the  rectangle  has  the  familiar  ratio  1.472.  The  diam- 
eter of  the  foot  has  the  ratio  .854.  If  an  .854  rectangle  is  constructed  in  the  centre 
of  the  containing  area,  and  a whirling  square  rectangle  applied  at  the  bottom,  the 
top  of  the  upper  painted  band  is  seen  to  be  at  half  the  height  of  the  whirling 
square  rectangle.  The  ratio  of  the  height  of  this  whirling  square  rectangle  is  .528. 
This  equals  1.000 -.472.  The  ratio  of  the  height  of  the  bowl  without  the  foot  is 
.944,  or  .472  X 2.  The  area  on  each  side  between  the  edge  of  the  foot  and  the  edge 
of  the  bowl  is  made  up  of  two  whirling  square  rectangles  (.309).  The  ratios  are: 

Height 1.000  Diameter  of  bowl 1.472 

Height  of  bowl  without  foot 944  Diameter  of  bottom  of  bowl . . . .708  (?) 

Width 2.0652  Diameter  of  foot 854 

In  view  of  the  small  size  of  these  two  Corinthian  skyphoi,  the  complicated  sys- 
tems of  proportion  which  they  reveal  are  probably  not  to  be  regarded  as  due  to 
conscious  design.  The  Attic  black-figured  and  red-figured  skyphoi,  on  the  other 
hand,  have  proportions  most  of  which  can  be  expressed  in  simple  ratios  and  by 
means  of  simple  geometrical  constructions. 

BLACK-FIGURED  PERIOD 

104  Large  Skyphos.  Inv.  20.18.  Robinson,  Catalogue,  no.  372.  Hambidge, 
p.  106.  (A)  Frieze  of  warriors  riding  on  dolphins.  (B)  Frieze  of  youths  riding  on 
ostriches. 

Height,  0.16  m.  Width,  including  handles,  0.298  m.  Greatest  diameter  of 
bowl,  0.221  m.  Smallest  diameter  of  bowl,  0.099  m.  Diameter  of  foot,  0.134  m. 

[ 149  ] 


GEOMETRY  OF  GREEK  VASES 


The  ratios  are: 

Height 1.000 

Width  including  handles 1.854  (error,  0.00128  m.) 

Greatest  diameter  of  bowl 1 .382 

Smallest  diameter  of  bowl 618 

Diameter  of  foot 854  (error,  0.00264  m.) 

The  proportions  are  clearly  shown  in  the  large  drawing,  and  the  simplicity  of 
the  scheme  appears  in  the  three  small  diagrams. 

This  skyphos  and  one  in  the  Metropolitan  Museum  furnish  one  of  the  rare 
instances  in  which  two  vessels  of  the  same  type  are  constructed  on  identically  the 
same  scheme  throughout,  except  for  the  disposition  of  the  painted  ornament.  In 
the  present  example  there  is  a very  slight  error  in  the  total  width,  and  a slightly 
greater  error  in  the  diameter  of  the  foot.  Cf.  above,  p.  29  and  Hambidge,  op. 
cit.,  where  drawings  of  both  skyphoi  are  published. 

105  Large  Skyphos.  Inv.  99.523.  Ann.  Rep.  1899,  p.  64,  no.  21.  (A) 
Mounted  Amazon  in  combat  with  an  egg-shaped  monster.  ( B ) Lion  attacking 
four  bulls. 

Height,  0.157  m.  Width,  0.295  m.  Diameter  of  bowl,  0.2205  m.  Diameter  of 
foot,  0.130  m. 


[ 150  ] 


SKYPHOS 


Without  the  handles  the  skyphos  is  contained  in  a y/2  rectangle.  The  ratios  are : 

Height 1.000 

Width 1.8787  = 1.00  + (.2929  X 3) 

Diameter  of  bowl 1.4142  = y/2 

Diameter  of  bottom  of  bowl 6143  = .2071  X 3 

Diameter  of  foot 8284  = .2071  X 4 


106  Large  Skyphos.  Inv.  99.524.  (A)  Amazon  holding  horse.  ( B ) Man 
holding  horse. 

Height,  0.168  m.  Width,  0.3005  m.  Diameter  of  bowl,  0.2255  m.  Diameter  of 
foot,  0.141  m. 

The  whole  vase  is  apparently  contained  in  four  y/b  rectangles,  the  ratio  being 
1.7888  = .4472X4;  and  the  bowl  is  contained  in  three  y/5  rectangles,  having  the 

[ 151  J 


GEOMETRY  OF  GREEK  VASES 


ratio  1.3416  = .4472X3.  The  diameter  of  the  bottom  of  the  bowl  has  the  ratio 
.618.  The  foot  apparently  has  the  ratio  .8416  = 1.3416  — .500. 


TABLE  OF  RATIOS  OF  THE  THREE  LARGE  SKYPHOI 


No. 

Height 

Width 

Diameter  of  bowl 

Diameter  of 
bottom  of  bowl 

Diameter  of 
foot 

104 

1.000 

1.854 

1.382 

.618 

.854 

105 

1.000 

1.8787 

1.4142 

.6143 

.8284 

106 

1.000 

1.7888 

1.3416 

.618 

.8416 

107 

107  Small  Skyphos.  Inv.  88.319.  No  figure  decoration. 

Height,  0.105  m.  Width,  including  handles,  0.188  m.  Diameter  of  bowl, 
0.1315  m.  Diameter  of  foot,  0.076  m. 

The  ratio  is  1.7888  ( = .4472X4),  four  \/5  rectangles.  The  ratio,  excluding  the 
handles  is  1.236,  i.  e.,  two  whirling  square  rectangles.  The  diameter  of  the  bottom 
of  the  bowl  is  one-third  of  the  total  width,  expressed  geometrically  in  the  drawing 
by  the  intersection  of  the  diagonal  of  the  whole  shape  with  the  diagonal  of  half  of 
the  shape.  The  ratio  of  the  height  to  the  diameter  of  the  foot  is  1.382.  If  a square 
be  subtracted  from  this  shape,  the  remainder,  .382  ( =2.618),  with  its  simple  sub- 
divisions, expresses  the  proportions  of  the  details  of  the  foot.  The  ratios  are: 

Height 1.000  Diameter  of  bottom  of  bowl 5963 

Width 1.7888  Diameter  of  foot 7236- 

Diameter  of  bowl 1 .236 

108  Small  Skyphos.  Inv.  97.373.  No  figure  decoration.  An  incised  line  near 
the  middle  of  the  bowl;  painted  bands  near  the  bottom. 

Height,  0.083  m.  Width,  0.1535  m.  Diameter  of  bowl,  0.093  m.  Diameter  of 
foot,  0.049  m. 


[ 152  ] 


SKYPHOS 


108 


The  enclosing  area  has  the  ratio,  1.854,  i.  e.,  it  is  composed  of  three  whirling 


square  rectangles.  The  ratio  of  the  bowl  is  1.118,  or 


V5 
2 ' 


The  proportions  of  de- 


tails can  all  be  accurately  expressed  in  terms  of  the  y/b  rectangles  composing  the 
area  of  the  bowl. 

The  ratios  are : 


Height 1.000  Diameter  of  bottom  of  bowl 500 

Width 1.854  Diameter  of  foot 591 

Diameter  of  bowl 1.118 


109  Small  Skyphos,  with  off-set  lip,  and  without  foot.  Inv.  76.48.  (A)  and 
( B ) Seilen  and  two  maenads. 


[ 153  ] 


GEOMETRY  OF  GREEK  VASES 


Height,  0.0845  m.  Width,  including  handles,  0.1555  m.  Greatest  diameter, 
0.104  m.  Diameter  of  base,  0.037  m. 

The  ratios  are : 


Height 

Width 

Greatest  diameter  of  bowl 

Diameter  of  base 


1.000 

1.854  = .618  X 3 
1.236  = .618  X 2 


.4472  = 


1 

2.236 


110  Skyphos  of  unusual  shape,  without  foot,  and  with  vertical  handles.  Inv. 

08.292.  Ann.  Rep.  1908,  p.  61.  Erotic  scenes. 

Height,  0.108-0.111  m.  Width,  0.187  m.  Diameter  of  bowl  at  top,  0.135  m., 
at  bottom,  0.102  m. 

The  enclosing  rectangle  apparently  has  the  ratio  1.691.  If  the  handles  are 
omitted  the  area  has  the  familiar  ratio,  1.236,  or  two  whirling  square  rectangles. 
The  ratios  are: 

Height 1.000  Diameter  of  bowl  at  top 1.236 

Width 1.691  Diameter  of  bowl  at  bottom 927 

RED-FIGURED  PERIOD 

111  Skyphos.  Inv.  76.49.  Hambidge,  p.  108.  Laurel  wreath  below  lip;  near 
bottom  a reserved  band  with  cross-hatching. 

112  Skyphos.  Inv.  01.8032.  Beazley,  V.  A.,  p.  130.  (A)  The  rising  of  Kore. 
( B ) Maenad  and  seilen.  “Penthesilea  Painter.” 

113  Skyphos.  Inv.  01.8076.  Beazley,  V.  A.,  p.  65.  Hambidge,  p.  109.  (A) 
Herakles  and  man.  ( B ) Man  addressing  youth. 

[ 154  ] 


SKYPHOS 


114  Skyphos.  Inv.  01.8097.  Beazley,  V.  A.,  p.  65.  (A)  Nestor  and  Euaichme. 
(B)  Aktor  and  Astyoche. 

115  Skyphos.  Inv.  10.176.  Beazley,  V.  A.,  p.  90.  Hambidge,  p.  111.  Ath- 
letes practising  the  jump.  “ Brygos  Painter.” 

116  Skyphos.  Unregistered.  Indecent. 

117  Skyphos.  Inv.  13.186.  Signed  by  Hieron  and  Makron.  Beazley,  V.  A., 
p.  101.  Hambidge,  p.  109.  The  seduction  and  return  of  Uelen. 

The  measurements  of  the  skyphoi  are  noted  in  Table  1.  Table  2 gives  the 
ratios  of  the  various  parts,  the  height  being  taken  as  unity  in  each  case. 

[ 155  ] 


GEOMETRY  OF  GREEK  VASES 


TABLE  1.  DIMENSIONS 


No. 

Height 

Width  including 
handles 

Greatest  diameter 
of  bowl 

Smallest  diameter 
of  bowl 

Diameter  of  foot 

Ill 

0.096  m. 

0.182  m. 

0.118  m. 

0.0715  m. 

0.0785  m. 

112 

0.228 

0.398 

0.279 

0.177 

0.194 

113 

0.183 

0.3245 

0.2255 

0.137 

0.1455 

114 

0.177 

0.308 

0.213 

0.132 

0.1415 

115 

0.145 

0.2605 

0.179 

0.109 

0.1225 

116 

0.0905 

0.161 

0.101 

0.065 

0.072 

117 

0.215 

0.389 

0.279 

0.159 

0.180 

[ 156  ] 


SKYPHOS 


TABLE  2.  RATIOS 


No. 

Height 

Width  including 
handles 

Greatest  diameter 
of  bowl 

Smallest  diameter 
of  bowl 

Diameter  of  foot 

Ill 

1.000 

1.8944 

1.236 

.764 

.809 

112 

1.000 

1.750(?) 

1.236 

.764(?) 

.854 

113 

1.000 

1.764 

1.236 

.764(?) 

.7888(?) 

114 

1.000 

1.750 

1.200 

.750 

.800 

115 

1.000 

1.809 

1.236 

.764 

.8396 

116 

1.000 

1.764 

1.118 

.736 

.809 

117 

1.000 

1.809 

1.309 

.736 

.8396 

[ 157  ] 


GEOMETRY  OF  GREEK  VASES 


The  most  striking  result  of  the  study  of  these  skyphoi  is  . the  reappearance  in 
four  of  them  of  the  familiar  ratio  1.236,  i.  e.,  two  whirling  square  rectangles,  as  the 
rectangle  enclosing  the  bowl.  The  same  shape  was  also  used  for  skyphoi  of  the 
black-figured  period,  as  we  have  seen  (nos.  107,  109,  110)  and  Mr.  Hambidge  has 
noted  several  other  examples  in  New  York  and  New  Haven  (op.  cit.,  pp.  110  ff., 
figs.  13, 14, 16, 17, 18  in  addition  to  the  skyphos  by  Hieron  in  the  British  Museum, 
The  Diagonal  I,  p.  114).  In  the  four  examples  here  published  the  projection  of 
the  handles  varies,  as  does  the  diameter  of  the  foot.  But  the  diminution  of  the 
diameter  of  the  bowl  is  practically  the  same  in  all  four,  the  lower  diameter  being 
expressed  by  the  ratio  .764  exactly  in  two  cases,  less  accurately  in  the  other 
two.  The  skyphos,  no.  4,  though  its  proportions  fall  in  the  “ static”  class,  being 
expressed  in  squares,  does  not  differ  noticeably  in  appearance  from  the  four  ex- 
amples just  mentioned.  The  last  two  skyphoi,  on  the  other  hand,  vary  strikingly 
from  what  may  perhaps  be  called  the  normal  shape.  No.  6,  a small,  but  beauti- 
fully made  skyphos,  is  slenderer  with  a slighter  diminution  from  top  to  bottom. 
No.  7,  the  famous  skyphos  signed  by  Hieron  and  Makron,  is  abnormal  in  its 
breadth,  in  the  excessive  diminution,  and  in  its  contours.  From  the  point  of  view 
of  form  it  is  perhaps  the  least  pleasing  of  the  seven.  The  shape  seems  to  have  been 
chosen  with  a view  to  affording  a more  advantageous  field  for  the  pictorial  com- 
position — a frieze  of  numerous  large,  closely  spaced  figures. 


C 158  ] 


TWO-HANDLED  CUP 


TWO-HANDLED  CUP 

118  Inv.  01.8023.  Ann.  Rep.  1901,  p.  34,  no.  23.  Impressed  decoration. 
A and  B,  Perseus  and  Medusa.  The  whole  surface  of  the  vase  is  covered  with 
black  varnish. 

Height,  0.117  m.  Width,  0.170  m.  Diameter  of  lip,  0.119  m.  Diameter  of 
bowl,  0.1175  m. 

The  over-all  ratio  is  1.4472,  i.  e.,  a square  plus  a s/5  rectangle.  If  the  handles 
are  omitted  the  enclosing  shape  is  very  close  to  a square.  The  proportional  rela- 
tion of  details  is  shown  in  the  drawing. 

The  ratios  are  as  follows : 

Height 1.000 

1.4472 
1.000 
1.000 

.8944  = .4472  X 2 
.500 

.5528  (inexact) 


Diameter  of  lip 

Diameter  of  bowl 

Diameter  of  shoulder 

Diameter  of  bottom  of  bowl 
Diameter  of  foot 


C 159  ] 


GEOMETRY  OF  GREEK  VASES 


KANTHAROS 

As  Professor  Tarbell  has  pointed  out  in  his  publication  of  no.  121,  this  type  of 
wine-cup  is  very  frequently  represented  in  vase  paintings  and  on  marble  reliefs, 
while  comparatively  few  actual  examples  of  the  best  period  have  survived.  This 
fact  suggests  that  kantharoi  were  generally  made  of  another  material  than  terra- 
cotta ; and  the  delicacy  of  handles  and  stems  leaves  little  doubt  that  this  material 
was  metal  — gold,  silver  or  bronze.  Of  the  five  kantharoi  here  published,  all, 
except  possibly  the  first,  appear  to  follow  closely  metal  prototypes.  It  happens 
that  three  of  them  are  signed,  nos.  119,  120  being  from  the  famous  pottery  of 
Nikosthenes,  no.  121  from  the  equally  famous  pottery  of  Hieron.  No.  122, 
though  unsigned,  is  in  all  probability  to  be  assigned  to  Brygos,  since  the  decora- 
tion is  by  the  “Brygos  Painter.”  The  name  HEN04>ANT0,  incised  on  the  bottom 
of  no.  123,  is  probably  that  of  the  owner  rather  than  the  maker.  Numbers  121, 
122,  123  are  among  the  most  interesting  examples  of  “dynamic  symmetry  ” in 
the  collection. 


119  Kantharos.  Inv.  00.334.  Beazley,  V.  A.,  p.  23.  (A)  Above,  Dionysos 
resting.  Below,  Herakles  and  the  lion.  ( B ) Above,  Sacrifice.  Below,  Herakles 
and  the  bull.  Signed  by  Nikosthenes  as  potter.  “ Painter  of  the  London  Sleep 
and  Death.” 

The  proportions  of  this  vase  cannot  be  determined,  because  the  weight  of  the 
handles  pulled  the  bowl  out  of  shape  before  the  firing.  The  lip  is  oval  instead  of 
circular,  and  the  enclosing  rectangle  is  lower  and  wider  than  was  intended.  In  the 
drawing  an  attempt  has  been  made  to  restore  the  original  appearance  by  increas- 
ing the  height  of  the  handles  by  4 mm.,  and  decreasing  the  width  proportionally. 
The  dimensions  thus  obtained  are:  height  of  handles,  0.237  m.  width,  0.294  m. 
This  rectangle  is  close  to  the  familiar  shape  1.236.  The  rectangle  in  which  the 
kantharos  without  its  handles  is  contained  is  also  nearly  of  this  shape. 

[ 160  ] 


KANTHAROS 


120  Kantharos.  Inv.  95.61.  Beazley,  V.  A.,  p.  23.  Erotic  scenes.  Signed  by 
Nikosthenes  as  potter.  “ Painter  of  the  London  Sleep  and  Death.” 

Height  of  one  handle,  0.238  m.,  of  the  other  0.235  m.  Average:  0.2365  m. 
Width  (handle  to  handle)  0.278  m.  Height  of  lip,  0.1635  m.  (average).  Diameter 
of  lip,  0.2025  m.  Diameter  of  foot,  0.1015  m. 

The  over -all  ratio  is  .854  (1.1708) ; and  the  ratio  of  the  rectangle  withthe  area 
above  the  lip  omitted  is  .5854  (1.708).  That  these  two  ratios  are  related  is  ap- 
parent, but  the  geometrical  analysis  is  complicated.  The  vase  without  the 
handles  is  enclosed  in  the  simple  rectangle  1.236  (.809).  This  shape  has  bee  n used 
as  the  basis  of  the  analysis.  It  can  be  seen  at  once  that  the  height  of  the  bowl 
(including  the  ring  at  the  top  of  the  stem)  equals  half  its  diameter,  and  that  the 
diameter  of  the  foot  also  equals  half  the  diameter  of  the  bowl.  The  upper  of  the 
two  horizontal  lines  near  the  bottom  of  the  bowl  is  at  half  the  height  of  the  lip. 
The  ratios,  regarding  the  height  of  the  bowl  as  unity,  are  as  follows: 


Height  to  lip 1 .000 

Height  of  upper  horizontal  line  near  bottom  of  bowl 500 

Height  of  bowl  without  stem 618 

Height  of  stem  to  bottom  of  ring .382 

Diameter  of  bowl 1.236 

Diameter  at  horizontal  line  near  bottom 708 

Diameter  of  stem  at  top 146 

Diameter  of  foot 618 


[ 161  ] 


GEOMETRY  OF  GREEK  VASES 


121  Kantharos.  Inv.  95.36.  Beazley,  V.  A.,  p.  90.  Tarbell,  University  of 
Chicago  Decennial  Publications,  6 (1902),  p.  3,  pi.  2-3.  Hambidge,  p.  68.  ( A ) 
Zeus  pursuing  Ganymede.  ( B ) Zeus  pursuing  a nymph.  By  the  Brygos  painter, 
and,  in  all  probability,  from  the  factory  of  Brygos. 

Height  to  top  of  unbroken  handle,  0.241  m.  Height  to  lip,  0.1675  m.  Greatest 
width  (handle  to  handle),  0.270  m.  Diameter  of  bowl,  0.1885  m.  Cf.  introduction, 
pp.  30  ff. 

The  total  width  divided  by  1.118  equals  0.24148,  which  is  within  half  a 

millimeter  of  the  height  of  the  unbroken  handle.  Every  detail  of  the  vase  is  clearly 
expressible  in  terms  of  the  rectangle  1.118.  If  the  area  above  the  lip  is  omitted, 
the  remainder  is  a whirling  square  rectangle.  If  the  area  below  the  bottom  of  the 
bowl  is  omitted,  the  remainder  is  again  a whirling  rectangle.  In  other  words  the 
total  area  may  be  regarded  as  two  overlapping  whirling  square  rectangles,  and  the 
bowl  is  contained  in  the  overlapping  portion.  If  a vertical  whirling  square  rec- 
tangle be  applied  at  either  end  it  defines  the  diameter  of  the  stem  at  its  junction 
with  the  body,  the  stem  being  contained  in  the  overlapping  portion.  The  diam- 
eter of  the  lip  is  fixed  by  the  intersection  of  the  diagonals  of  half  the  shape  with 
the  horizontal  line  at  the  level  of  the  lip.  This  shows  that  the  main  proportions  of 
the  kantharos  without  its  handles  are  the  same  as  those  of  the  kantharos  with  its 
handles.  The  diameter  of  the  ridge  near  the  bottom  of  the  bowl  equals  half  the 
total  height.  It  coincides  with  the  central  square  of  each  of  the  two  \/ 5 rectangles 

[ 162  ] 


KANTHAROS 


of  which  the  area  is  composed.  The  diameter  of  the  foot  is  obtained  from  the 
intersection  of  two  whirling  square  rectangle  diagonals.  It  is  noteworthy  that, 
as  in  the  kantharos  no.  122,  the  diameter  of  the  stem  at  its  junction  with  the 
body  is  also  determined  by  the  intersection  of  the  diagonals  of  the  large  whirling 
square  rectangle  applied  at  the  bottom  of  the  total  area.  These  points  can  also 
be  determined  in  a third  manner.  If  a semicircle  be  described  with  the  centre  of 
the  lip  as  centre  and  the  radius  of  the  lip  as  radius  its  intersections  with  the  cen- 
tral square  of  the  lower  V 5 rectangle  determine  both  the  height  and  the  diameter 
of  the  ridge  near  the  bottom  of  the  bowl,  and  its  intersections  with  the  sides  of  the 
two  overlapping  vertical  whirling  square  rectangles  determine  the  height  and 
diameter  of  the  top  of  the  stem.  The  bowl  is  practically  contained  in  this  semi- 
circle (cf.  the  semicircle  in  no.  120). 

The  following  table  of  proportions  gives  the  ratios,  (a)  with  the  height  re- 
garded as  unity,  (6)  with  the  width  regarded  as  unity. 


A B 

Height  to  top  of  handles 1.000  .8944 

Projection  of  handles  above  lip 309  .2764 

Height  of  lip 691  .618 

Height  of  upper  line  near  bottom  of  bowl 4045  .3618 

Height  of  lower  line  near  bottom  of  bowl 3944  .3527 

Height  of  stem 309  .2764 

Width 1.118  1.000 

Diameter  of  bowl 7725  .691 

Diameter  of  ridge  near  bottom  of  bowl 500  .4472 

Diameter  of  stem  at  junction  with  bowl 118  .1055 

Diameter  of  foot 4045  .3618 


122  Kantharos.  Inv.  98.932.  Signed  by  Hieron  as  potter.  Gigantomachy, 
Pollack,  Zwei  Vasen  aus  der  Werkstatt  Hierons,  p.  28,  pi.  4,  5.  Beazley,  V.  A., 

p.  109. 

The  vase  has  been  put  together  from  many  fragments,  and  some  pieces  are 
missing,  especially  the  greater  portion  of  the  bottom  of  the  bowl  and  the  upper 
part  of  one  handle.  But  the  stem  is  complete,  and  enough  of  the  bottom  of  the 
bowl  is  preserved  to  give  almost  its  entire  contour;  and  the  incomplete  handle  has 
been  accurately  restored  to  match  the  other.  Under  these  circumstances  an  in- 
vestigation of  the  proportions  seemed  feasible. 

The  height  of  the  complete  handle  is  0.262  m.  The  greatest  width  (handle  to 
handle)  is  0.277  m.  These  dimensions  make  a rectangle  of  the  shape  1.0557,  more 
easily  intelligible  in  terms  of  its  reciprocal,  .9472,  i.  e.,  500+.4472  or  half  a square 
plus  a y/5  rectangle;  and  the  geometrical  scheme,  though  necessarily  complicated 
because  of  the  numerous  elements  to  be  accounted  for,  is  clear  and  consistent 
throughout. 

The  three  small  diagrams  will  help  to  make  this  scheme  intelligible.  In  the 
first  of  these  diagrams  the  V 5 rectangle  has  been  placed  above  the  two  squares. 
The  lip  is  seen  to  be  at  half  the  height  of  the  y/ 5 rectangle.  Perpendiculars  from 

[ 163  ] 


GEOMETRY  OF  GREEK  VASES 


the  centres  of  the  two  squares  give  the  diameter  of  the  upper  of  the  two  hori- 
zontal lines  near  the  bottom  of  the  bowl.  This  diameter  is,  therefore,  one-half  of 
the  total  width.  The  diameter  of  the  foot  is  one-third  of  the  total  width,  and  can 
be  determined  geometrically  by  the  intersections  of  diagonals  of  each  of  the  two 
squares  with  the  diagonal  of  the  rectangle  composed  of  both  the  squares.  If  two 
diagonals  of  half  the  over-all  shape  are  drawn  from  the  top  of  the  vertical  axis  to 
the  lower  corners  their  intersection  with  diagonals  of  the  squares  determine  the 
diameter  of  the  lip. 

In  the  second  small  diagram  a whirling  square  rectangle  has  been  applied  at 
the  bottom  of  the  over-all  shape.  By  applying  the  reciprocals,  and  the  reciprocals 
of  the  reciprocals,  the  simple  figure  is  obtained  which  is  illustrated  by  diagram  IX, 
page  7.  The  upper  of  the  two  horizontal  lines  near  the  bottom  of  the  bowl  coin- 
cides with  the  horizontal  division  in  this  figure.  The  greatest  diameter  of  the 


KANTHAROS 


lower  part  of  the  bowl  is  determined  by  the  intersections  of  the  diagonals  of  the 
applied  squares  with  this  line.  The  diameter  of  the  lip  can  also  be  determined  by 
the  intersections  of  the  diagonal  of  half  the  over -all  shape  with  the  top  of  the 
whirling  square  rectangle.  The  diagonals  of  the  whirling  square  rectangle  fix  the 
diameter  of  the  top  of  the  stem. 

In  the  third  diagram  the  area  above  the  lip  has  been  omitted.  The  remaining 
rectangle  is  easily  seen  to  be  made  up  of  two  squares  plus  two  \/5  rectangles, 
i.  e.,  .7236,  or  1.382.  The  intersections  of  the  diagonals  of  half  this  shape  with  the 
upper  of  the  two  lines  near  the  bottom  of  the  bowl,  determine  a smaller  1.382  rec- 
tangle, the  width  of  which  is  equal  to  the  diameter  of  the  lower  of  the  two  lines 
near  the  bottom  of  the  bowl  (the  ridge).  If  squares  be  applied  at  either  end  of  this 
rectangle  their  centres  determine  both  the  height  and  the  diameter  of  the  ridge  in 
the  stem.  The  intersection  of  their  diagonals  at  the  axis  of  the  vase  fixes  the 
height  of  the  stem. 

The  kantharos  without  its  handles  is  enclosed  in  the  rectangle  .9045,  i.  e., 
.500+ .4045.  The  reciprocal  of  .9045  is  1.1056,  which  also  can  be  divided  into 
familiar  ratios  such  as  .5528X2,  or  .382+. 7236. 

In  the  following  table  are  given:  (A)  the  ratios,  ( B ) the  dimensions  computed 
on  the  assumption  that  the  total  width  is  0.2764  m.,  (C)  the  actual  dimensions  as 
measured  by  the  writer,  (D)  the  dimensions  published  by  Pollack,  op.  cit.  The 
total  width  is  taken  as  .2764  m.,  instead  of  0.277  m.,  because  it  happens  to  coin- 
cide with  a familiar  ratio.  The  greatest  variation  between  the  figures  in  columns 
B and  C is  less  than  a millimetre. 


Height  to  top  of  handles 

Height  of  lip 

Height  of  upper  line  near  bottom  of  bowl. . . . 
Height  of  lower  line  near  bottom  of  bowl 

Height  of  stem 

Height  of  ridge  in  stem 

Total  width 

Diameter  of  lip 

Diameter  of  upper  line  near  bottom  of  bowl  . . 
Diameter  of  lower  line  near  bottom  of  bowl  . 

Diameter  of  stem  at  top  and  at  ridge 

Diameter  of  foot 


A 

B 

c 

D 

.9472 

0.2618  m 

0.262  m 

0.26  m 

.7236 

0.200 

0.2005 

0.20 

.382 

0.1056 

0.106 

.354 

0.0978 

0.0975 

.264 

0.07295 

0.073 

.191 

0.528 

0.053 

1.000 

0.2764 

0.277 

.6584 

0.1819 

0.181 

0.18 

.500 

0.1382 

0.1375 

.528 

0.1459 

0.145 

.1459 

0.0403 

0.0405 

.333 

0.092 

0.0925 

0.091 

123  Kantharos.  Inv.  01.8081.  Hambidge,  p.  123.  No  figure  decoration. 
The  whole  surface,  except  for  one  member  of  the  base,  covered  with  a fine  black 
varnish.  The  vase  is  intact,  and  of  extremely  careful  workmanship. 

Height  of  handles,  0.1535  m.  Height  of  bowl,  0.111m.,  width  (handles), 
0.222  m.  Diameter  of  bowl,  0.146  m.  Diameter  of  base,  0.0995  m. 

[ 165  ] 


GEOMETRY  OF  GREEK  VASES 


This  kantharos  fits  very  accurately  the  familiar  shape  composed  of  a square 
and  a y/5  rectangle  — 1.4472,  or  .691.  If  the  projection  of  the  handles  above  the 
lip  is  omitted  the  rectangle  is  composed  of  two  squares. 

The  ratio  of  the  diameter  of  the  base  to  the  total  width  is  .4472:1.000,  that  is 
if  a \/5  rectangle  is  applied  at  the  bottom  of  the  over-all  rectangle,  the  diameter 
of  the  base  of  the  kantharos  coincides  with  the  side  of  the  central  square  of  the 
y/b  rectangle.  All  the  details  are  accounted  for  by  simple  subdivisions  of  the 
over-all  rectangle  or  the  applied  y/b  rectangle.  The  proportioning  of  the  mould- 
ings of  the  base  is  noteworthy,  each  member  being  twice  as  wide  as  the  member 
immediately  below  it. 

The  proportions  of  details  are  given  below  in  two  forms,  the  width  being  taken 
as  unity  in  the  first  column,  the  height  in  the  second. 


Height  of  handles 691  1.000 

Height  of  lip 500  .7236 

Height  of  ridge  near  bottom  of  bowl 1236  .1788 

Width 1.000  1.4472 

Diameter  of  bowl 6584  .9528 

Diameter  of  ridge  near  bottom  of  bowl.  . 5236  .7577 

Smallest  diameter 382  .5367 

Diameter  of  foot 4472  .6472 


C 166  ] 


KYLIX 


KYLIX 

BLACK-FIGURED  PERIOD 
(A)  Kylix  with  deep  bowl , off-set  lip,  and  high  stem. 

Of  the  eight  examples  here  published  two  bear  the  signature  of  the  potter 
Tleson,  son  of  Nearchos,  three  that  of  Xenokles,  one  that  of  Hermogenes.  Their 
shapes,  the  thinness  of  their  walls,  the  delicacy  of  handles  and  feet  suggest  that 
they  follow  metal  prototypes.  The  painted  decoration  on  the  exterior  is  limited 
to  palmettes  adjoining  the  handles,  and  in  some  examples  a small  design  on  each 
side  of  the  lip.  The  drawing  of  the  kylix  by  Tleson,  no.  124,  shows  which  portions 
are  normally  left  in  the  color  of  the  clay  and  which  are  covered  with  black  varnish. 
It  is  noteworthy  that  the  bottom  of  the  foot  curves  up  slightly  at  the  edge  to 
prevent  breaking  when  the  cup  is  set  down.  The  kylix,  no.  124,  is  proportioned 
on  the  basis  of  the\/2  rectangle.  No.  128  works  out  in  squares.  The  proportions 
of  no.  126  seem  to  be  related  to  the  \/3  rectangle.  The  remaining  five  exhibit 
familiar  shapes  derived  from  the  rectangle  of  the  whirling  squares.  For  conveni- 
ence of  comparison  the  four  most  important  ratios  are  given  here  in  tabular  form : 


Height 

Width 

Diameter 
of  bowl 

Diameter 
of  foot 

124 

Tleson 

1.000 

2.1213 

1.5988 

.7071 

125 

Hermogenes 

1.000 

2.0652 

1.5652 

.6708 

126 

Xenokles 

1.000 

2.0206 

1.4433 

.7217 

127 

Xenokles 

1.000 

2.045 

1.427 

128 

Unsigned 

1.000 

2.000 

1.5000 

.750 

129 

Tleson 

1.000 

2.000 

1.4472 

.6666? 

130 

Unsigned 

1.000 

1.882 

1.382 

.5647 

131 

Xenokles 

1.000 

1.854 

1.382 

.6753 

[ 167  ] 


GEOMETRY  OF  GREEK  VASES 


124  Kylix.  Inv.  98.920.  Hambidge,  p.  53.  Signed  by  Tleson,  son  of  Nearchos, 
as  potter.  Interior  picture,  a wounded  stag. 

Average  height,  0.14235  m.  Width  (handle  to  handle),  0.30  m.  Diameter  of 
bowl,  0.227  m.  Diameter  of  foot,  0.10  m. 

The  enclosing  area  is  made  up  of  three  \/2  rectangles  placed  vertically  side  by 
side,  the  ratio  being  2.1213,  or  .7071 X 3.  It  can  readily  be  seen  that  if  the  width 
of  the  over -all  area  is  taken  as  30  cm.,  the  width  of  each  of  the  \/2  rectangles  is 
10  cm.,  and  their  height  is  14.142  cm.  This  comes  within  a millimetre  of  the 
average  height  obtained  from  eight  measurements  (.1425,  141,  142,  1435,  142, 
1435,  143,  1415).  If  a square  be  applied  at  the  bottom  of  the  three  y/2  rectangles 
the  intersection  of  its  diagonal  with  the  diagonal  of  the  y/2  rectangle  determines 
the  height  of  the  stem.  The  diameter  of  the  bowl  is  determined  by  the  intersec- 
tion of  the  diagonal  of  the  area  which  is  in  excess  of  the  applied  square  with  the 

diagonal  of  half  the  -y/2  rectangle. 

The  ratios  are : 

Height 1.000  Width 2.1213 

Height  of  bowl 5858  Diameter  of  bowl 1.5988 

Height  of  stem 4142  Diameter  of  foot 7071 


[ 168  ] 


KYLIX 


125  Kylix.  Iny.  95.17.  Signed  by  Hermogenes  as  potter.  On  each  side  of  lip, 
a hen. 

Height,  0.1425  m.  Width,  0.2895  m.  Diameter  of  bowl,  0.220  m.  Diameter  of 
foot,  0.094  m. 

The  enclosing  area  is  the  rectangle  2.0652  ( = 1.000+.618+.4472).  If  the 
projection  of  the  handles  is  omitted  the  remaining  rectangle  has  the  ratio  1.5652, 
or  2,0652  — .500.  This  rectangle  can  also  be  regarded  as  1.118+.4472.  If  .4472  is 
cut  off  from  each  end  the  remainder  is  .6708.  This  is  the  ratio  of  the  diameter  of 
the  foot.  The  height  of  the  stem  is  .4472. 

The  ratios  are : 


Height 

. ...  1.000 

Height  of  lip  alone 

1708 

Width 

. ...  2.0652 

Height  of  bowl  alone 

382 

Diameter  of  bowl 

. ...  1.5652 

Height  of  stem 

......  .4472 

Height  of  bowl  with  lip 

5528 

Diameter  of  foot 

6708 

C 169  ] 


GEOMETRY  OF  GREEK  VASES 


126  Kylix.  Inv.  99.529.  Signed  by  Xenokles  as  potter.  Lip  black.  On  re- 
served band  below  lip,  the  signatures  between  palmettes. 

Height,  0.141  m.  Width,  0.284.  Diameter  of  lip,  0.207  m.  Diameter  of  foot, 
0.101  m. 

This  vase  is  apparently  composed  on  a \/3  theme.  The  over-all  area  is  a \/3 
rectangle  plus  one-sixth  of  a \/3  rectangle.  The  ratio  is  2.02065.  No  simple  ratio 
can  be  obtained  for  the  diameter  of  the  bowl,  but  if  the  diameter  at  the  top  and 
bottom  of  the  moulding  which  forms  the  upper  member  of  the  lip  be  taken  instead 
(as  is  done  in  the  drawing)  the  ratio  is  1.4433,  or  a \/3  rectangle  minus  one-sixth 
of  a \/3  rectangle.  The  diameter  of  the  foot  is  half  this  diameter.  The  kylix 
may  be  regarded  as  composed  within  two  overlapping  \/3  rectangles,  the  bowl 
being  placed  in  the  overlapping  portion.  The  intersection  of  diagonals  of  these 
\/3  rectangles  fixes  the  height  of  the  junction  of  the  bowl  and  stem.  The  height 
ratio  of  the  bowl  is  .58337.  If  twice  this  portion  be  subtracted  from  the  ratio  of 
the  diameter  of  the  bowl,  the  remainder,  .2766  is  the  ratio  of  the  diameter  of  the 
ring  at  the  top  of  the  stem.  This  is  expressed  geometrically  in  the  drawing  by 
applying  squares  at  either  end  of  the  rectangle  enclosing  the  bowl. 

The  ratios  are : 


Height 

Width 

Diameter  of  bowl  (excluding  moulding  of  lip) 

Projection  of  each  handle 

Height  of  bowl 

Height  of  stem 

Diameter  of  ring  at  top  of  stem 

Diameter  of  foot 


1.000 

2.0206  - V3  + 

6 


1.4433  = 


.2886  - 

.58337 

.41663 

.2766 

.7217 


V3  - 
V3 
6 


V3 

6 


[ 170  ] 


KYLIX 


127  Kylix.  Inv.  98.921.  Signed  by  Xenokles  as  potter.  Interior,  forepart  of 
horse  and  rider.  One  handle  missing. 

Height,  0.092  m.  Diameter  of  bowl,  0.132  m.  Width,  including  handles 
(estimated)  0.188  m. 

The  over-all  rectangle  appears  to  be  2.045,  the  rectangle  with  the  handles 
omitted  being  1.427.  The  projection  of  each  handle  is  .309  or  .618-^2.  The  pro- 
portions of  details  cannot  be  expressed  in  simple  subdivisions  of  these  shapes. 
The  diameter  of  the  foot  is  perhaps  to  be  obtained  from  the  intersection  of  the 
diagonal  of  the  rectangle  1.736  (2.045  — .309)  with  the  diagonal  of  the  rectangle 
1.0225  (2.045-7-2). 


[ 171  ] 


GEOMETRY  OF  GREEK  VASES 


128  Kylix.  Inv.  89.268.  Lip  black.  On  the  reserved  band  below  the  lip, 
seilens  pursuing  maenads. 

Height,  0.0946  m.  Width,  0.1895  m.  Diameter  of  bowl,  0.141  m.  Diameter  of 
foot,  0.0705  m. 

The  over-all  area  is  composed  of  two  squares,  and  most  of  the  details  work  out 
simply  in  squares. 


Height 1 Projection  of  each  handle 

Width 2 Diameter  of  stem 

Diameter  of  lip 1|  Height  of  lip 


Diameter  of  foot 


[ 172  ] 


KYLIX 


129  Kylix.  Inv.  92.2655.  Signed  by  Tleson,  son  of  Nearchos,  as  potter. 

Height,  0.1335  m.  Width,  0.267  m.  Diameter  of  bowl,  0.1925  m.  Diameter  of 
foot,  0.09  m. 

With  the  handles  the  kylix  is  enclosed  in  a\/4  rectangle,  i.  e.,  two  squares. 
Without  the  handles  it  is  enclosed  in  a 1.4472  rectangle,  i.  e.,  a square  plus  a \/5 
rectangle.  The  diameter  of  the  foot  is  nearly  one-third  the  width.  The  height  of 
the  lip,  and  the  diameter  of  the  stem  are  each  equal  to  one-fifth  the  height  of  the 
vase.  The  bowl  without  handles  or  stem  fits  a 2.764  rectangle.  The  geometrical 
analysis  is  omitted,  since  no  simple  method  of  expressing  these  ratios  has  been 
found. 


130  Kylix.  Inv.  92.2654.  Hambidge,  p.  117.  Meaningless  lettering.  On 
each  side  of  the  lip,  a swan. 

Height,  0.148  m.  Width,  0.281  m.  Diameter  of  bowl,  0.204  m.  Diameter  of 
foot,  0.0865  m. 

The  area  in  which  the  kylix  without  its  handles  is  enclosed  has  the  familiar 
ratio  1.382.  With  th.Q  handles  added  the  rectangle  has  nearly  the  shape,  1.882, 
or  1. 382+. 500.  The  diameter  of  the  foot  is  apparently  determined  by  the  inter- 
section of  the  diagonal  of  half  the  1.382  rectangle  with  the  diagonal  of  the  ap- 
plied square.  The  ratio  of  this  diameter  is  .5647.  The  height  of  the  stem  is  .427, 
that  of  the  bowl  .528.  If  the  bowl  without  stem  or  handles  is  considered  sepa- 
rately, it  is  seen  to  conform  to  the  shape,  2.618  (=  .382);  and  the  height  of  the 
bowl  to  the  junction  with  the  lip  is  to  the  height  of  the  lip  as  1.000  is  to  .618.  Cf. 
Mr.  Hambidge’s  analysis,  which  brings  out  some  further  points. 


[ 173  ] 


GEOMETRY  OF  GREEK  VASES 


131  Kylix.  Inv.  95.18.  Signed  by  Xenokles  as  potter.  Interior,  a sphinx. 
Exterior,  (A)  Two  centaurs.  ( B ) Lion  and  faun. 

Height,  0.109  m.  Width,  0.2025  m.  Diameter  of  bowl,  0.153  m.  Diameter  of 
foot,  0.0735  m. 

The  enclosing  area  is  1.854,  or  three  whirling  square  rectangles.  Without  the 
handles  the  vase  is  contained  in  a 1.382  rectangle.  The  main  proportions  are 
thus  exactly  those  of  the  black-figured  skyphos,  no.  104.  The  shape  can  also  be 
considered  as  two  horizontal  whirling  square  rectangles  overlapping,  the  over- 
lapping portion  being  the  rectangle  of  the  bowl.  The  intersection  of  the  diagonal 
one  of  these  whirling  square  rectangles  with  the  diagonal  of  half  the  over-all  area 
determines  the  diameter  of  the  foot. 

The  ratios  are : 


Height 1.000 

Width 1.854 

Diameter  of  lip 1.382 

Diameter  of  bowl  at  junction  with  lip  . 1 .236 


Diameter  of  foot 6753 

Height  of  stem 472 

Height  of  bowl  and  lip 528 


[ 174  ] 


KYLIX 


(B)  Eye  Kylix,  with  shallower  bowl,  and  shorter  and  heavier  stem. 

This  type  of  kylix,  apparently  adapted  by  the  Athenian  potters  of  the  late 
black-figured  period  from  an  Ionian  type,  became  the  favorite  shape  in  the  red- 
figured  period,  in  which  it  underwent  further  modifications.  All  three  of  the 
examples  here  published  have  large  pairs  of  eyes  painted  on  them  in  the  black- 
figured  technique.  The  remaining  painted  decoration  is  black-figured  on  no.  132, 
red-figured  on  nos.  133  and  134. 


132  Kylix.  Inv.  03.784.  Hambidge,  p.  116.  The  painted  decoration,  which  is 
confined  to  the  exterior,  is  in  the  black-figured  technique.  On  each  side,  between 
two  large  eyes,  a seilen  grasping  vines.  Beneath  each  handle,  a siren.  Below,  a 
frieze  of  lions,  pursuing  pegasi  and  a deer. 

Height,  0.1205  m.  Width,  0.346  m.  Diameter  of  bowl,  0.274  m.  Diameter  of 
foot,  0.121  m. 

If  the  height  is  taken  as  0.1212  m.,  the  over-all  ratio  is,  2.854.  The  ratio  of  the 
bowl  is  fairly  close  to  2.236,  i.  e.,  a y/h  rectangle.  The  error  is  3 mm.  The  diam- 


eter of  the  foot  is  equal  to  the  height.  The  junction  of  bowl  and  stem  is  deter- 
mined by  the  intersection  of  the  diagonals  of  the  two  overlapping  whirling  square 
rectangles  in  the  -y/5  rectangle. 

The  ratios  are : 


Height 

1.000 

Diameter  of  bowl 

2.236  + 

Height  of  stem 

309 

Projection  of  each  handle 

.309- 

Height  of  bowl  alone 

691 

Diameter  of  ring  at  top  of  stem . 

.500 

Width 

2.854 

Diameter  of  foot 

1.000 

[ 175  ] 


GEOMETRY  OF  GREEK  VASES 


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Height 1.000  Projection  of  each  handle 309 

Height  of  stem 4331  Diameter  of  ring  at  top  of  stem 500 

Height  of  bowl  alone 5669  Smallest  diameter  of  stem 4045 

Width 3.236  Diameter  of  foot 1.000 

Diameter  of  bowl 2.618 


KYLIX 


1 

J 


C 177  ] 


width  by  .0025917  m.  The  geometrical  analysis  is  very  simple.  Cf.  the  small  diagram.  The  ratios  are: 

Height 1.000  Width 3.0606  Smallest  diameter  of  stem ..  .3535 

Height  of  stem  to  junction  with  bowl  .4571  Diameter  of  bowl 2.3535  Diameter  of  top  of  stem  . . . .4142 

Height  of  bowl  alone 5429  Projection  of  each  handle.  .3535  Diameter  of  foot  9393 


GEOMETRY  OF  GREEK  VASES 


RED-FIGURED  PERIOD 


The  collection  is  especially  rich  in  examples  of  kylikes  of  the  archaic  period. 
A number  of  them  are,  however,  so  incomplete  as  to  make  the  analysis  of  their 
proportions  impossible,  or  at  least  unreliable.  A few  others  have  been  omitted 
because  they  seemed  to  fit  no  geometric  scheme  convincingly.  The  examples 
chosen  for  publication  are  fairly  representative.  They  fall  into  two  main  classes, 
the  kylix  with  a fairly  tall,  slender  stem,  nos.  135-160,  and  the  stemless  kylix  found 
in  the  period  of  the  free  style,  nos.  161,  162.  Examination  of  the  table  of  ratios 
shows  that  certain  simple  ratios  occur  repeatedly.  The  total  width  of  the  kylikes 
with  stems  varies  between  3.000  and  3.618.  The  two  most  common  widths  are 
3.236,  or  two  whirling  square  rectangles,  and  3.4142,  or  two  squares  plus  a \/2 
rectangle.  The  commonest  ratios  for  the  diameter  of  the  bowl  are  2.618  ( = .382) 


and  2.7071,  or  2 + 


V2 
2 ' 


The  former  occurs  four  times,  three  times  in  connection 


with  the  over-all  ratio  3.236.  The  latter  is  found  three  times  in  connection  with 
the  shape  3.4142.  The  diameter  of  the  foot  varies  between  .927  and  1.236.  In 
eleven  examples  out  of  twenty-six  it  is  exactly,  or  almost  exactly,  equal  to  the 


height  of  the  kylix.  The  projection  of  the  handles  is  .309,  or  in  six  cases.  The 

diameter  of  the  raised  ring  on  the  foot  has  the  ratio  .691  in  six  cases.  If  one  were 
to  choose  an  ideal  scheme  it  might  well  be  the  following : 


Width 3.236 

Diameter  of  bowl 2.618 

Diameter  of  stem 236 


Diameter  of  ring  on  foot 691 

Diameter  of  foot 1.000 

Projection  of  handles 309 


The  level  of  the  junction  between  bowl  and  stem  cannot  be  measured  exactly. 
It  will  be  found  to  be  at  about  half  the  height  in  the  majority  of  large  kylikes. 
The  two  kylikes,  nos.  146,  147  conform  to  this  ideal  scheme  in  their  main  propor- 
tions. The  former  happens  to  be  signed  by  the  potter  Pamphaios.  Nos.  140,  151, 
155  are  also  perhaps  from  his  factory,  though  this  cannot  be  proved.  They  are 
among  the  best  examples  of  dynamic  symmetry  in  the  series.  It  is  noteworthy 
that  two  of  the  kylikes  decorated  by  the  “ Panaitios  Painter”  who  worked  in 
the  factory  of  Euphronios  have  identically  the  same  proportions  throughout 
(nos.  159,  160). 

Of  the  two  stemless  kylikes  one  conforms  to  a simple  scheme  based  on  the 
whirling  square  rectangle,  while  the  other  belongs  to  the  static  class. 

The  ratios,  so  far  as  they  have  been  determined,  are  given  in  the  following 
table : 


[ 178  ] 


KYLIX 


No. 

Height 

Width 

Diameter  of 
bowl 

Diameter  of 
stem 

Diameter  of 
ring  on  foot 

Diameter  of 
foot 

Projection  of 
each  handle 

135 

1.000 

3.000 

2.191 

.4472 

1.000 

.4045 

136 

1.000 

3.000 

2.3504 

.5669 

1.0512 

.3248 

137 

1.000 

3.000 

2.382 

.1708? 

.691 

.8944 

.309 

138 

1.000 

3.090 

2.236 

.309 

1.000 

.427 

139 

1.000 

3.090 

2.382 

.5609? 

.927? 

.354 

140 

1.000 

3.090 

2.472 

.236 

.5652 

.8944- 

.309 

141 

1.000 

3.236 

2.472 

.691 

1.000- 

.382 

142 

1.000 

3.236 

2.528 

.236 

.764 

1.000 

.354 

143 

1.000 

3.236 

2.528 

.236 

.691 

1.146 

.354 

144 

1.000 

3.236 

2.545 

.236- 

.691 

1.000 

.3455 

145 

1.000 

3.236 

2.618 

.764 

.927 

.309 

146 

1.000 

3.236 

2.618 

.236 

.691 

1.000- 

.309 

147 

1.000 

3.236 

2.618 

.236 

.618 

1.000+ 

.309 

148 

1.000 

3.4142 

2.4714 

.4714 

1.0572 

.4714 

149 

1.000 

3.4142 

2.5606 

.2677 

1.0606 

.4268 

150 

1.000 

3.4142 

2.5858 

1.000 

.4142 

151 

1.000 

3.4142 

2.7071 

.2929 

? 

.3535 

152 

1.000 

3.4142 

2.7071 

.7071 

1.0606 

.3535 

153 

1.000 

? 

? 

? 

? 

? 

? 

154 

1.000 

(3.4142) 

2.7071 

.5469 

1.000 

(.3535) 

155 

1.000 

3.382 

2.618 

.309 

.545 

1.000 

.382 

156 

1.000 

3.382 

2.764 

.236 

.764 

1.000 

.309 

157 

1.000 

3.528 

2.764 

.764 

1.090? 

.382 

158 

1.000 

3.618 

2.764 

.666 

1.236 

.427 

159 

1.000 

3.618 

2.854 

1.146 

.382 

160 

1.000 

3.618 

2.854 

.7236 

1.146 

.382 

161 

1.000 

3.854 

2.854 

1.545 

.500 

162 

1.000 

4.666 

3.500 

2.000 

.5833 

[ 179  ] 


GEOMETRY  OF  GREEK  VASES 


135  Kylix.  Inv.  00.336.  Off-set  lip.  Beazley,  V.  A.,  p.  22.  Interior,  Youth 
testing  arrow.  Early  archaic  style. 

Height,  0.07975  m.  Width,  including  handles,  0.238  m.  Diameter  of  bowl, 
0.173  m.  Diameter  of  foot,  0.079  m. 

The  enclosing  rectangle  is  obviously  composed  of  three  squares,  and  the  foot 
is  exactly  contained  in  the  central  square.  Without  the  handles  the  kylix  is  placed 
in  a 2.191  rectangle,  the  projection  of  the  handles  being  .4045,  or  four  whirling 
square  rectangles.  The  diameter  of  the  stem  is  .4472;  its  height,  including  the 
ring,  .250.  The  height  of  the  lip  is  .236. 

136  Kylix.  Inv.  10.179.  Beazley,  V.  A.,  p.  86,  no.  19.  Interior,  Seilen 
seated  on  a pithos.  “ Panaitios  Painter.” 

Height,  0.10635  m.  Width,  including  handles,  0.318  m.  Diameter  of  bowl, 
0.248  m.  Diameter  of  foot,  0.11  m. 

The  containing  area  is  composed  of  three  squares,  but  the  analysis  of  details  is 
unusually  complicated.  If  a whirling  rectangle  be  applied  at  the  bottom  of  the 
central  square,  the  intersection  of  the  diagonal  of  half  this  rectangle  with  the 
diagonal  of  the  reciprocal  fixes  the  diameter  of  the  ring  on  the  foot,  as  is  shown 
by  the  perpendicular  dropped  from  the  point  of  the  intersection.  The  length  of 


this  perpendicular  is  .3504  (the  reciprocal  of  2.854).  The  diameter  of  the  foot  is 
.3504X3,  or  1.0512.  The  diameter  of  the  bowl  is,  2.3504,  i.  e.,  two  squares  plus  the 
ratio  .3504.  If  a horizontal  line  is  drawn  at  the  height  .3504,  the  diagonals  of 
each  half  of  the  containing  shape  cut  the  line  so  as  to  fix  the  diameter  of  the  foot. 
The  rectangle  with  the  diameter  of  the  foot  as  width  and  the  .3504  line  as  height 
is  thus  shown  to  be  made  up,  like  the  over-all  rectangle,  of  three  squares.  In  the 
drawing  the  diagonal  of  the  small  square  at  the  right  is  continued,  and  cuts  the 
top  of  the  over-all  rectangle  so  as  to  define  the  diameter  of  the  bowl.  The  pro- 
jection of  the  handles  beyond  the  bowl  is  expressed  by  the  ratio  .3248,  which  is 
the  reciprocal  of  3.0787  — a rectangle  made  up  of  two  squares  and  a 1.0787  (.927) 
rectangle  composed  of  one  and  a half  whirling  square  rectangles.  It  is  noteworthy 
that  the  top  of  the  1.0787  rectangle  coincides  with  the  line  at  the  level  .3504. 
The  scheme  is  more  clearly  shown  in  the  small  diagram. 

It  is  also  noteworthy  that  if  the  diameter  of  the  foot  be  taken  as  unity  the 
total  width  and  the  diameter  of  the  bowl  are  expressible  in  simple  ratios,  viz., 


.3168 

.111 


2.854  and 


.2483 

.111 


= 2.236  = V5. 


C 180  ] 


KYLIX 


[ 181  ] 


GEOMETRY  OF  GREEK  VASES 


137  Kylix.  Inv.  89.272.  Beazley,  V.  A.,  p.  105,  no.  69.  Interior,  Man  and 
boy.  Exterior,  Men  and  youths.  Attributed  to  Makron. 

Height,  0.14  m.  Width,  including  handles,  0.422  in.  Diameter  of  bowl, 
0.332  m.  Diameter  of  foot,  0.124  m. 

The  ratios  are : 


With  handles 3.000.  0.14  X 3 =.420.  Error,  0.002  m. 

Without  handles ...  2.382.  0.14  X 2.382  =.33348.  Error,  0.0015  m. 

Dihmeter  of  foot 8944.  0.14  X .8944  = .126216.  Error,  0.0022  m. 

Diameter  of  ring  on  foot 691 

Diameter  of  stem 1708? 

Projection  of  handles 309 


138  Kylix.  Inv.  01.8074.  Off -set  lip.  Interior,  Crouching  archer. 

Height,  0.0736  m.  Width,  including  handles,  0.228  m.  Diameter  of  bowl, 
0.1645  m.  Diameter  of  foot,  0.075  m. 

If  the  height  is  made  0.074  m.  (0.0006  m.,  more  than  the  average;  0.001  m., 
less  than  the  diameter  of  the  foot)  the  ratios  are : 


With  handles 3.09.  0.074  X 3.09  = .22866.  Error,  0.00066  m. 

Without  handles 2.236.  0.074  X 2.236  = .165464.  Error,  0.000964  m. 

Foot 1.000.  0.074  X 1.000  = .074.  Error,  0.001  m. 

Diameter  of  stem 309 

Height  of  lip 236 

Projection  of  handles 427 


[ 182  ] 


KYLIX 


[ 183  ] 


GEOMETRY  OF  GREEK  VASES 

139  Kylix.  Inv.  13.84.  Beazley,  V.  A.,  p.  132.  Interior,  Youth  and  woman. 

Exterior  (A)  and  ( B ),  Seilens  and  maenads.  “Penthesilea  Painter.” 

Height,  0.1146  m.  Width, ./including  handles,  0.351  m.  Diameter  of  bowl, 
0.271  m.  Diameter  of  foot,  0-105  m. 

Assuming  the  height  to  be  0.114  m.,  the  ratios  are: 


With  handles 3.090.  0.114  X 3.090  = .35226.  Error,  0.001226  m. 

Without  handles 2.382.  0.114  }<  2.382  = .271548.  Error,  0.000548  m. 

Diameter  of  foot 927? 

Diameter  of  ring  on  foot .5609? 

Projection  of  handles 354 


140  Kylix.  Inv.  95.35.  Beazley,  V.  A.,  p.  24.  Hambidge,  p.  119.  Interiof, 
Athlete  with  jumping  weights  and  javelins.  Exterior  ( A ) and  (B),  Seilens  and 
maenads.  The  paintings  are  placed  by  Beazley  in  the  group  headed  by  the  works 
of  the  ‘ ‘ Painter  of  the  London  Sleep  and  Death.”  The  potter  may  well  have  been 
Pamphaios. 

Height,  0.1337  m.  Width,  including  handles,  0.413  m.  Diameter  of  bowl, 
0.3315  m.  Diameter  of  foot,  0.1195  m. 

The  ratios  are : 


With  handles 

. . 3.090. 

0.1337X3.090  =.413133. 

Error,  0.000133  m. 

Without  handles 

. . 2.472. 

0.1337X2.472  =.3394964. 

Error,  0.001094  m. 

Diameter  of  foot 

. . .8944 

0.1337  X .8944  = . 1337. 

Error,  0.00008  m. 

Diameter  of  ring  on  foot . . 

. . .5652 

Diameter  of  stem 

. . .236 

Projection  of  handles 

. . .309 

1 

[ 184  ] 


KYLIX 


C 185  ] 


GEOMETRY  OF  GREEK  VASES 


141  Kylix.  Inv.  01.8029.  Beazley,  V.  A.,  p.  98.  Interior,  Youth  at  laver. 
Unsigned;  the  painting  is  ascribed  to  Douris. 

Height,  0.07925  m.  Width,  including  handles,  0.2615  m.  Diameter  of  bowl, 
0.198  m.  Diameter  of  foot,  0.077  m. 

If  the  height  be  assumed  to  be  0.08  m.,  the  ratios  are: 


With  handles 3.236.  0.08  X 3.236  = .25888.  Error,  0.00262  m. 

Without  handles 2.472.  0.08  X 2.472  = .19776.  Error,  0.00024  m. 

Diameter  of  foot 1.000?  0.08  X 1.000  =0.08.  Error,  0.003  m. 

Diameter  of  ring  on  foot 691 

Projection  of  handles 382 


142  Kylix.  Inv.  00.338.  Beazley,  V.  A.,  p.  97,  no.  6.  Interior,  Diskoboles. 
Exterior  (A)  and  (B),  Battle  scenes.  Signed  by  Douris  as  painter. 

Height,  0.09325  m.  Width,  including  handles,  0.3055  m.  Diameter  of  bowl, 
0.2395  m.  Diameter  of  foot,  0.095  m. 

Assuming  the  height  to  be  0.094  m.,  the  ratios  are: 


With  handles 3.236.  0.094  X 3.236  = .304184.  Error,  0.00132  m. 

Without  handles 2.528.  0.094  X 2.528  = .237632.  Error,  0.00187  m. 

Diameter  of  foot 1.000.  Error,  0.001  m. 

Diameter  of  ring  on  foot 764 

Diameter  of  stem 236 

Projection  of  handles 354 


[ 186  ] 


KYLIX 


[ 187  ] 


GEOMETRY  OF  GREEK  VASES 


143  Kylix.  Inv.  01.8020.  Beazley,  V.  A.,  p.  86,  no.  8.  Interior,  Diskobolos. 
Exterior,  Athlete.  “ Panaitios  Painter.” 

Height,  0.08893  m.  Width,  including  handles,  0.291  m.  Diameter  of  bowl, 
0.224  m.  Diameter  of  foot,  0.10175  m. 

If  the  height  is  taken  as  0.0895  m.  (0.00057  m.,  greater  than  the  average). 
The  ratios  are : 


With  handles 3.236.  0.0895  X 3.236  = .289622.  Error,  0.00138  m. 

Without  handles 2.528.  0.0895  X 2.528  = .226256.  Error,  0.0022  m. 

Foot 1.146.  0.0895  X 1.146  = .102567.  Error,  0.00081  m. 

Smallest  diameter  of  stem 236 

Diameter  of  ring  on  foot 691 

Projection  of  handles 354 


144  Kylix.  Inv.  01.8034.  Beazley,  V.  A.,  p.  93.  Interior,  Symposion.  Ex- 
terior, Symposion.  “ Berlin  Foundry  Painter.” 

Height,  0.1172  m.  Width,  including  handles,  0.3755  m.  Diameter  of  bowl, 
0.298  m.  Diameter  of  foot,  0.1175  m. 

Assuming  the  height  to  be  0.117  m.,  the  ratios  are: 


With  handles 3.236.  0.117  X 3.236  = .378612.  Error,  0.003112  m. 

Without  handles 2.545.  0.117  X 2.545  = .297765.  Error,  0.00023  m. 

Diameter  of  foot 1.000.  Error,  0.0005  m. 

Diameter  of  ring  on  foot 691 

Diameter  of  stem 236  — 

Projection  of  handles 3455  = .691  -f-  2 


[ 188  ] 


KYLIX 


[ 189  ] 


GEOMETRY  OF  GREEK  VASES 

145  Kylix.  Inv.  01.8022.  Beazley,  V.  A.,  p.  105,  no.  70.  Hambidge,  p.  121. 
Interior,  Woman  and  man.  Exterior,  Youths,  men  and  women.  Attributed  to 
Makron. 

Height,  0.126  m.  Width,  including  handles,  0.413  m.  Diameter  of  bowl, 
0.332  m.  Diameter  of  foot,  0.1175  m. 

Assuming  the  height  of  the  kylix  to  be  0.127  m.,  the  ratios  are: 


With  handles 3.236.  0.127  X 3.236  - .410972.  Error,  0.002028  m. 

Without  handles 2.618.  0.127  X 2.618  = .332486.  Error,  0.000486  m. 

Diameter  of  foot 927.  0.127  X .927  = .117729.  Error,  0.000229  m. 

Diameter  of  ring  on  foot ...  .764 

Projection  of  handles 309 


146  Kylix.  Inv.  95.32.  Beazley,  V.  A.,  p.  23.  Interior,  Seilen.  Exterior  (A) 
Arming  scene.  (B)  Combat.  Signed  by  Pamphaios  as  potter. 

Height,  0.1324  m.  Width,  including  handles,  0.425  m.  Diameter  of  bowl, 
0.3435  m.  Diameter  of  foot,  0.129  m. 

If  the  height  be  assumed  to  be  0.1315  m.,  the  ratios  are: 


With  handles 3.236.  0.1315  X 3.236  = .425534.  Error,  0.000534  m. 

Without  handles 2.618.  0.1315  X 2.618  = .343267.  Error,  0.000233  m. 

Diameter  of  foot.  . 1.000.  Error,  0.0025  m. 

Diameter  of  ring  on  foot 691 


Height  of  top  of  painted  band  on  exterior,  beneath  figures 618 

Diameter  of  top  of  painted  band 1.618 

Diameter  of  bottom  of  painted  band , 1.309 


Diameter  of  stem 236 

Projection  of  handles 309 


[ 190  ] 


KYLIX 


[ 191  ] 


GEOMETRY  OF  GREEK  VASES 


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[ 192  ] 


Without  handles 2.618.  0.119  X 2.618  - .311542.  Error,  0.000542  m. 

Diameter  of  foot 1.000  — ? Error,  0.004  m.  (?) 

Diameter  of  ring  on  foot 618 

Diameter  of  stem 236 

Projection  of  handles 309 


KYLIX 


148  Kylix.  Inv.  95.33.  Beazley,  V.  A.,  p.  14.  Signed  by  Chachrylion  as  pot- 
ter. Interior,  Maenad  with  castanets.  “ Hermaios  Painter.” 

Height,  0.0765  m.  Width,  including  handles,  0.263  m.  Diameter  of  bowl, 
0.192  m.  Diameter  of  foot,  0.082  m, 

If  the  height  be  taken  as  .077  m.,  the  proportions  fit  very  accurately  a scheme 
based  on  the  y/2  rectangle.  The  containing  area  is  made  up  of  two  squares  plus  a 
y/2  rectangle  (3.4142).  The  bowl  is  enclosed  in  the  rectangle  2.4714,  or  two 
squares  plus  one-third  of  a y/2  rectangle.  The  projection  of  the  handles  beyond 


■\/2 

the  bowl  is  again  .4714  or  The  scheme  is  most  clearly  comprehended  when 

O 

^/2  \/2  y/2 

the  total  area  is  regarded  thus:  — - 4-  1 + - + 1 + Cf.  the  accompanying 

O O O 

diagram. 

The  central  rectangle  fixes  the  diameter  of  the  stem;  and  simple  subdivi- 

O 


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the  ratio  1.0572,  or  2.4714  — 1.4142  ^2  + — V2,  or  2 — • The  projection 

of  the  bowl  beyond  the  foot  is  .7071 
The  ratios  are : 


Height 

Width 

Diameter  of  bowl . . . 
Diameter  of  foot.  . . . 
Diameter  of  stem . . . 
Projection  of  handles 


1.000 

3.4142  = 2 + V2 
V2 
3 

2a/2 


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1.0572  = 2 - 


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O 


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V2 


C 193  ] 


GEOMETRY  OF  GREEK  VASES 


[ 194  ] 


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[ 195  ] 


GEOMETRY  OF  GREEK  VASES 


[ 196  ] 


ming  the  height  to  be  0.915  m.,  the  ratios  are: 

With  handles 3.4142.  0.915  X 3.4142  = .3123992.  Error,  0.00039993  m. 

Without  handles 2.5858.  0.915  X 2.5858  = .2366.  Error,  0.0021  m. 

Diameter  of  foot 1.000 

Projection  of  handles 4142 


KYLIX 


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[ 197  ] 


Without  handles 2.7071.  0.1125  X 2.7071  = .304548.  Error,  0.000045  m. 

Diameter  of  stem 2929  (=  reciprocal  of  3.4142) 

Projection  of  handles 3535 

The  diameter  of  the  foot  is  apparently  not  expressible  in  a simple  ratio.  In  the  drawing  it  is  obtained 
by  a complicated  geometrical  construction  which  is  not  convincing. 


GEOMETRY  OF  GREEK  VASES 


152  Kylix.  Inv.  00.344.  Beazley,  V.  A.,  p.  189.  Interior,  Herakles,  Nessos, 
Deianeira.  Exterior,  Centauromachy.  Signed  by  Erginos  as  potter,  and  by 
Aristophanes  as  painter. 


153  Kylix.  Inv.  00.345.  Beazley,  V.  A.,  p.  189.  The  paintings  are  a replica 
of  those  on  the  above  kylix,  but  are  unsigned. 

These  two  cups  are  a pair.  Though  only  one  is  signed,  both  are  decorated  by 
the  same  hand,  and  both  must  have  been  made  in  the  same  factory.  One  might 
expect  that  they  would  conform  to  the  same  scheme.  This,  however,  is  not  the 
case.  The  unsigned  example  fits  fairly  accurately  a simple  scheme.  The  signed 
piece  varies  widely  in  the  height,  the  total  width,  and  the  diameter  of  the  bowl. 
The  discrepancy  in  the  diameter  of  the  foot  and  the  diameter  of  the  ring  on  the 
foot,  though  much  smaller,  is  still  appreciable.  Attempts  to  fit  the  signed  vase 
into  another  scheme  have  remained  unsuccessful.  The  dimensions  of  the  two 
cups  are  as  follows: 


No.  152 

No.  153 

Height 

. . . . 0.1329  m. 

0.1315  m. 

Width  including  handles 

. ...  0.442 

0.452 

Diameter  of  bowl 

. ...  0.34925 

0.3559 

Diameter  of  foot 

. ...  0.142 

0.1405 

Diameter  of  ring  on  foot 

. ...  0.096 

0.0925 

Projection  of  handles 

. ...  0.046375 

0.04855 

Diameter  of  maeander  band  on  exterior, 

A 

. . . 0.1345 

0.1335 

Diameter  of  maeander  band  on  exterior, 

B 

. . . 0.159 

0.159 

Diameter  of  maeander  band  on  interior, 

A 

. ...  0.163 

0.177 

Diameter  of  maeander  band  on  interior, 

B 

. . . . 0.192 

0.2085 

The  proportions  of  the  unsigned  kylix  are  as  follows : 


With  handles 

3.4142. 

0.1315 

X 

3.4142 

Without  handles 

2.7071. 

0.1315 

X 

2.7071 

Diameter  of  foot 

1.0606. 

0.1315 

X 

1.0606 

Diameter  of  ring  on  foot 

.7071 

Projection  of  handles. . . . 

.3535 

.4489673.  Error,  0.003  m. 
.355984.  Error,  0.0000984  m. 
.1394689.  Error,  0.001  m. 


[ 198  ] 


KYLIX 


l 199  ] 


GEOMETRY  OF  GREEK  VASES 


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[ 201  ] 


GEOMETRY  OF  GREEK  VASES 


156  Kylix.  Inv.  1353.15.  Lent  to  the  Museum  by  Professor  J.  C.  Hoppin. 
Beazley,  V.  A.,  p.  199.  Interior,  Seilen  and  maenad.  Exterior  (A)  and  ( B ) 
Seilens  and  maenads.  Attributed  to  Douris. 

Height,  0.1075  m.  Width,  including  handles,  0.359  m.  Diameter  of  bowl, 
0.294  m.  Diameter  of  foot,  0.1065  m. 

If  the  height  is  made  equal  to  the  diameter  of  the  foot,  i.  e.,  0.1065  m.,  the 
ratios  are : 


With  handles 3.382.  0.1065  X 3.382  = .360183.  Error,  0.001183  m. 

Without  handles 2.764.  0.1065  X 3.382  = .294366.  Error,  0.000366  m. 

Diameter  of  foot 1.000 

Diameter  of  ring  on  foot 764 

Diameter  of  stem 236 

Projection  of  handles 309 


157  Kylix.  Inv.  01.8038.  Beazley,  V.  A.,  p.  91.  Interior,  Athlete  with 
strigil,  and  dog.  Attributed  to  the  Brygos  Painter. 

Height,  0.07443  m.  Width,  including  handle,  0.265  m.  Diameter  of  bowl, 
0.2065  m.  Diameter  of  foot,  0.0825  m. 

Assuming  the  height  to  be  0.075  m.,  the  ratios  are: 


With  handles 3.528.  0.075  X 3.528  = .2646.  Error,  0.0004  m. 

Without  handles 2.764.  0.075  X 2.764  = .2053.  Error,  0.0012  m. 

Diameter  of  foot 1.090? 

Diameter  of  ring  on  foot 764 

Projection  of  handles 382 


[ 202  ] 


KYLIX 


[ 203  ] 


GEOMETRY  OF  GREEK  VASES 


158  Kylix.  Inv.  01.8021.  Beazley,  V.  A.,  p.  86,  no.  6.  Interior,  Athlete  with 
rope.  Exterior,  Athletes.  “Panaitios  Painter.” 

Height,  0.0839  m.  Width,  including  handles,  0.302  m.  Diameter  of  bowl, 
0.2294  m.  Diameter  of  foot,  0.102  m. 

If  the  height  is  made  0.0834  m.,  i.  e.,  0.0005  m.,  less  than  the  average,  the 
ratios  are  as  follows: 


With  handles 3.618.  0.0834  X 3.618  = .30174.  Error,  0.00026  m. 

Without  handles 2.764.  0.0834  X 2.764  = .230517.  Error,  0.0011  m. 

Foot 1.236.  0.0834  X 1.236  = .10308.  Error,  0.00108  m. 

Ring  on  foot 666 

Projection  of  handles 427 


159  Kylix.  Inv.  98.876.  Beazley,  V.  A.,  p.  87.  Interior,  Youth  with  jumping 
weights.  Exterior,  Athlete  with  javelins  and  jumping  weights.  “ Panaitios 
Painter.” 

Height,  0.0821  m.  Width,  including  handles,  0.299  m.  Diameter  of  bowl, 
0.233  m.  Diameter  of  foot,  0.095  m. 


Ratio,  with  handles 3.618.  .0821  X 3.618  = .2970378.  Error,  2 mm. 

Ratio,  without  handles 2.854.  .0821  X 2.854  = .2343134.  Error,  1 mm. 

Ratio  of  foot 1.146.  .0821  X 1.146  = .0940866.  Error,  1mm. 


It  is  noteworthy  that  the  diameter  of  the  ring  at  the  top  of  the  foot  is  .382,  that 
the  diameter  of  the  foot  is  .382X3  and  that  the  projection  of  the  handles  beyond 
the  bowl  is  .382. 


[ 204  ] 


KYLIX 


[ 205  ] 


GEOMETRY  OF  GREEK  VASES 


160  Kylix.  Inv.  01.8018.  Beazley,  V.  A.,  p.  87.  Interior,  Symposion. 
“ Panaitios  Painter.” 

Height,  0.0884  m.  Width,  including  handles,  0.3155  m.  Diameter  of  bowl, 
0.249  m.  Diameter  of  foot,  0.1005  m. 

The  height  varies  between  0.0855  m.,  and  0.091  m.,  the  dimension  given  above 
being  the  average  obtained  from  eight  measurements.  If  the  height  is  reduced  by 
1 mm.,  the  ratios  are  as  follows: 


With  handles 3.618.  0.0875  X 3.618  = .3165.  Error,  0.001  m. 

Without  handles 2.854.  0.0875  X 2.854  = .2497.  Error,  0.0007  m. 

Foot 1.146.  0.0875  X 1.146  = .100275.  Error,  0.0002  m. 


STEMLESS  KYLIX 

Of  the  two  examples  investigated  the  first  shows  proportions  based  on  the 
whirling  square  rectangle,  the  second  belongs  to  the  “static”  class. 

161  Stemless  Kylix.  Inv.  01.8089.  Hambidge,  p.  120.  Exterior  (A)  and 
(. B ),  Combat  between  two  hoplites.  Free  style. 

Height,  0.074  m.  Width,  0.2845  m.  Diameter  of  bowl,  0.21  m.  Diameter  of 
foot,  0.114  m. 

The  details  fit  a simple  geometrical  scheme  very  exactly. 

The  ratios  are : 

Height 1.000 

Height  of  off-set  lip 2764 

Width 3.854 


Diameter  of  bowl 2.854 

Diameter  of  foot 1.545 

Projection  of  handles 500 


[ 206  ] 


STEMLESS  KYLIX 


[ 207  ] 


GEOMETRY  OF  GREEK  VASES 


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£ § 
p ^ 
t-i I 03 

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w 

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bfi 

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[ 208  J 


Height 1.000 

Width 4.066  = 4f 

Diameter  of  bowl 3.500  = 3| 

Diameter  of  foot 2.000  = 2 (inaccurate) 

Projection  of  handles 5833  = f 


LEKYTHOS 


LEKYTHOS 

BLACK-FIGURED  PERIOD 

163  Lekythos  of  Corinthian  Shape. 

Inv.  08.291.  Ann.  Rep.  1908,  p.  61. 

Pictures  in  three  zones. 

Height,  0.2095  m.  Diameter, 

0.093  m. 

This  vase  is  contained,  with  re- 
markable exactness,  within  a \/5  rec- 
tangle, and  its  details,  including  most 
of  the  painted  bands  are  clearly  de- 
finable in  terms  of  this  rectangle. 

A \/5  rectangle  may  be  divided 
into  five  reciprocals,  in  this  case  five 
\/5  rectangles  placed  horizontally. 

The  top  of  the  lowest  of  these  \/5 
rectangles  coincides  with  the  band  on 
which  the  lowest  of. the  three  zones  of 
decoration  rests.  The  diameter  of  the 
bottom  of  the  body  equals  the  height 
of  this  \/5  rectangle  (.4472),  or  the 
side  of  its  central  square.  Simple  sub- 
divisions of  the  adjoining  whirling 
square  rectangles  give  the  diameter  of 
the  foot  (.7888).  If  the  diagonals  of 
these  whirling  square  rectangles  are 
produced  they  meet  at  the  level  of 
the  band  on  which  the  second  zone 
of  decoration  rests,  i.  e.,  at  the  height 
of  two  whirling  square  rectangles 
placed  side  by  side  (.809). 

The  reciprocal  rectangle  applied  to 
the  top  of  the  containing  shape  ac- 
counts for  all  the  details  of  the  lip  and 
neck,  and  its  base  coincides  with  a 
painted  band  on  the  shoulder.  The 
diameter  of  the  lip  is  seen  to  equal  the  side  of  the  central  square  (.4472), 
and  all  the  details  of  lip  and  neck  can  be  simply  expressed  in  terms  of  this 
square.  The  diameter  of  the  neck  equals  the  side  of  a y/5  rectangle  placed 
in  the  middle  of  the  square.  This  \/5  rectangle  (.200)  is  a reciprocal  of  the 
reciprocal  of  the  over-all  shape.  The  upper  of  the  two  bands  on  which  the 
highest  of  the  three  zones  of  painted  decoration  rests  is  placed  at  the  level  at 

[ 209  ] 


163 


GEOMETRY  OF  GREEK  VASES 


which  the  diagonal  of  the  whole 
shape  intersects  the  diagonal  of 
a whirling  square  rectangle.  The 
area  above  this  line  is  thus  seen 
to  be  the  familiar  rectangle, 
1.0652. 

164  Lekythos.  Inv.  95.15. 
Hambidge,  p.  132.  Achilles  and 
Ajax  playing  draughts  in  the 
presence  of  Athena. 

Height,  0.2825  m.  Diameter, 
0.142  m. 

The  containing  rectangle  is 
composed  of  two  squares;  and 
all  the  proportions  except  the 
height  to  the  top  of  the  shoulder, 
the  height  of  the  neck,  and  its 
lower  diameter,  are  expressible 
in  terms  of  squares,  or  in  simple 
fractions,  using  the  diameter  as 
unity. 

The  ratios  are : 


Height 2.000  = 2 

Height  to  shoulder  . 1.400  = if 
Height  of  lip,  neck 

and  shoulder 600  = f 

Height  of  lip 1666  = £ 

Diameter 1.000  = 1 

Diameter  of  lip 500  = \ 

Smallest  diameter  of 

neck 200  = \ 

Diameter  of  bottom 

of  body 4286  = ? 

Diameter  of  foot  . . . .600  = l 


[ 210  ] 


LEKYTHOS 


165 


165  Lekythos.  Inv.  98.922.  Female  figure  mounting  a quadriga;  behind  her, 
Hermes. 

Height,  0.1105  m.  Diameter,  0.0582  m. 

The  containing  area  has  the  ratio,  1.8944;  it  is  composed  of  a square  plus  two 
V5  rectangles  (1.000+.4472+.4472).  The  proportions  of  details  fall  into  simple 


ratios,  as  follows : 

Height . 

1.8944 

Diameter 

1.000 

Height  to  shoulder 

1.2764- 

Diameter  of  lip 

.500 

Height  of  lip,  neck,  and  shoulder  .... 

.618+ 

Diameter  of  top  of  neck. . . . 

.250 

Height  of  lip  and  neck 

.500 

Diameter  of  bottom  of  body . 

.4472 

Height  of  painted  band  below  figures 

.500 

Diameter  of  foot 

.6584 

[ 211  ] 


GEOMETRY  OF  GREEK  VASES 


166 


166  Lekythos.  Inv.  99.528.  Warrior  lead- 
ing horse. 

Height,  0.2615  m.  Diameter,  0.09  m. 

The  containing  rectangle  has  the  ratio, 
2.8944,  or  1.4472X2,  i.  e.,  it  is  composed  of 
two  squares  plus  two  y/5  rectangles. 


The  ratios  are : 

Height 2.8944 

Height  to  neck 2.0854 

Height  of  neck  and  lip 809 

Height  of  lip 382 

Height  of  neck 427 

Diameter 1.000 

Diameter  of  lip 5669+ 

Dfa,meter  of  top  of  neck 236 

Diameter  of  bottom  of  neck 382 

Diameter  of  bottom  of  body 382 

Diameter  of  foot 764 


[ 212  ] 


LEKYTHOS 


RED-FIGURED  PERIOD 

167  Lekythos.  Inv.  13.195.  Beazley, 
V.  A.,  p.  26.  Hambidge,  p.  131.  Cows 
led  to  the  sacrifice.  Signed  by  Gales  as 
potter. 

Height,  0.310  m.  Diameter,  0.124  m. 

The  containing  area  is  made  up  of  two 
squares  and  a half.  All  the  proportions 
can  be  expressed  in  simple  fractions. 
The  lekythos  is  a good  example  of  what 
is  called  by  Mr.  Hambidge  “ static 
symmetry.” 

The  ratios  are: 

Height 2 1 

Height  to  bottom  of  shoulder  ....  If 

Height  of  lip,  neck  and  shoulder  . . f 

Height  of  lip w 

Height  of  neck s 

Height  of  shoulder |j 

Diameter 1 

Diameter  of  lip 1 

Diameter  of  bottom  of  body % 

Diameter  of  foot f 


[ 213  ] 


GEOMETRY  OF  GREEK  VASES 


168  Lekythos.  Inv.  13.199.  Bearded  man  in  Ionic  dress  reciting  and  playing 
lyre. 

Height,  0.3925  m.  Diameter,  0.14  m. 

169  Lekythos.  Inv.  13.198.  Beazley,  V.  A.,  p.  114.  Young  hunter  with  dog. 
“ Pan  Painter.” 

Height,  0.3875  m.  Diameter,  0.138  m. 

These  two  lekythoi  are  said  to  have  been  found  together,  and  resemble  one 
another  closely.  Both  are  contained  in  the  rectangle,  2.809.  The  proportions  of 

C 214  ] 


LEKYTHOS 


the  foot  are  the  same  in  both,  but  those  of  the 
lip,  neck  and  shoulder  are  different. 

The  ratios  are : 

No.  168  No.  109 


Height 2.809  2.809 

Height  to  shoulder 2.000  1.941? 

Height  of  lip,  neck  and  shoulder  .809  .808? 

Height  of  shoulder 2230  .250? 

Height  of  lip  and  neck 5854  .018 

Diameter 1 .000  1 .000 

Diameter  of  lip 5854  .5609 

Diameter  of  top  of  shoulder  . . . .236  .236 

Diameter  of  bottom  of  body  . . . .382  .382 

Diameter  of  foot 691  .091 


170  Lekythos.  Inv.  95.44.  Beazley,  V.  A ., 
p.  76.  Menelaos  and  Helen.  “ Providence 
Painter.” 

Height,  0.411  m.  Diameter,  0.142  m. 

The  enclosing  rectangle  has  the  ratio, 
2.8944.  The  height  to  the  shoulder  equals 
twice  the  diameter.  The  diameters  of  the  lip 
and  of  the  top  of  the  shoulder  are  obtained 
from  a whirling  square  rectangle  applied  at 
the  top  of  the  enclosing  area.  The  diameter 
of  the  bottom  of  the  body  and  of  the  foot 
are  expressed  in  terms  of  a whirling  rectangle 
applied  to  half  of  the  lower  large  square. 


Height  2.8944 

Height  to  shoulder  2.000 

Height  of  lip 3455 

Diameter 1 .000 

Diameter  of  lip 528 

Diameter  of  top  of  shoulder 236 

Diameter  of  bottom  of  body 382 

Diameter  of  boot 091 


[ 215  ] 


GEOMETRY  OF  GREEK  VASES 


171  Lekythos.  Inv.  13.189.  Beazley,  V.  A.,  p.  91.  Woman  taking  skein  of 
wool  out  of  a basket.  “ Brygos  Painter.” 

Height,  0.332  m.  Diameter,  0.1125  m. 

The  containing  rectangle  has  the  ratio  2.944  = 1.472X2.  The  vase  up  to  the 


shoulder  is  enclosed  in  two 
simple  ratios,  as  follows : 

squares. 

Most  of  the  details  can  be  expressed  in 

Height 

. ..  2.944 

Diameter 

. . 1.000 

Height  to  shoulder 

. ..  2.000 

Diameter  of  lip 

. . .5669 

Height  of  lip 

. . . .382 

Diameter  of  top  of  neck 

. . .236 

Height  of  neck 

. . . .309 

Diameter  of  bottom  of  body . . . 

. . .382 

Height  of  shoulder 

. . . .253 

Diameter  of  foot 

. . .666 

172  Lekythos.  Inv.  95.41.  Beazley,  V.  A.,  p.  98.  Athlete  with  jumping 
weights.  By  Douris. 

Height,  0.300  m.  Diameter,  0.101  m. 

The  containing  rectangle  is  composed  of  three  squares  (3.000).  The  diameter 
of  the  bottom  of  the  body  has  the  ratio  .400,  that  of  the  foot,  .800,  suggesting 
that  the  proportions  belong  to  the  static  class.  No  other  simple  ratios  have  been 
noted. 


[ 216  ] 


LEKYTHOS 


[ 217  ] 


GEOMETRY  OF  GREEK  VASES 


173  Lekythos.  Inv.  95.45.  Beazley,  V.  A.,  p.  76.  Apollo  with  phiale  and 
lyre.  “ Providence  Painter.” 

Height,  0.391  m.  Diameter,  0.1305  m. 

The  enclosing  area,  composed  of  three  squares,  is  divided  “ dynamically.” 
The  height  of  the  vase  to  the  neck  is  expressible  in  a simple  ratio.  The  height  of 
the  lip  (.3618  = 2.764)  makes  with  the  over -all  diameter  an  area  composed  of 
two  1.382  rectangle  (1.382X2  = 2.764).  The  intersection  of  the  diagonal  of  one 
of  these  rectangles  with  the  diagonal  of  a square  applied  to  it  fixes  the  diameter  of 
the  lip  (.5854).  The  diameters  of  the  bottom  of  the  body  (.382)  and  of  the  foot 
(.691)  are  found  often  in  lekythoi. 


Height 

3.000 

Diameter 

. . 1.000 

Height  to  neck 

2.309 

Diameter  of  lip 

. . .5854 

Height  to  shoulder 

2.059 

Diameter  of  top  of  neck 

. . .236 

Height  of  lip 

.3618 

Diameter  of  top  of  shoulder . . . , 

. . .2764 

Height  of  lip  and  neck 

.691 

Diameter  of  bottom  of  body . . . 

. . .382 

Height  of  shoulder 

.250 

Diameter  of  foot 

. . .691 

174  Lekythos.  Inv.  98.885. 

Seilen  pursuing  a maenad.  The  outlines  of  the 

figures  incised;  the  woman  painted  white. 

Height,  0.221  m.  Diameter,  0.0745  m. 

The  containing  rectangle  is 

composed  of  three  squares.  The  body  occupies 

two  squares;  the  lip,  neck,  and  shoulder  are  enclosed  in  the  remaining  square. 

The  details  are  expressible  in  simple  ratios  as  follows : 

Height 

3.000 

Diameter 

. ..  1.000 

Height  to  shoulder 

2.000 

Diameter  of  lip 

. . . .618 

Height  of  lip  and  shoulder 

1.000 

Diameter  of  top  of  neck 

. . . .236 

Height  of  lip 

.382 

Diameter  of  top  of  shoulder . . . 

. . . .382 

Height  of  neck 

.472 

Diameter  of  bottom  of  bodv . . . 

. . . .382 

Height  of  shoulder 

.146 

Diameter  of  foot 

. . . .764 

Height  of  foot 

.191 

[ 218  ] 


LEKYTHOS 


174 


C 219  ] 


GEOMETRY  OF  GREEK  VASES 


175  Lekythos.  Inv.  93.103.  Youth  and  woman  at  stele. 

Height,  0.4215  m.  Diameter,  0.1275  m. 

176  Lekythos.  Inv.  93.104.  Woman  with  perfume  vase  and  youth  with 
spear  at  stele. 

Height,  0.4215  m.  Diameter,  0.123  m. 

These  lekythoi  are  undoubtedly  a pair,  made  in  the  same  factory.  The  paint- 
ings are  by  a single  artist,  “a  florid  imitator  of  the  Achilles  Painter  ” (Beazley, 
Vases  in  America , p.  166).  They  are  of  the  same  height,  the  heights  of  all  the 
members  are  the  same;  the  diameters  are  also  the  same  except  for  the  largest 
diameter  of  the  body,  which  is  0.0035  m.,  less  in  the  second  than  in  the  first.  The 
first  can  be  simply  analyzed  in  terms  of  the  rectangle,  3.309,  as  appears  from  the 
drawing  and  the  table  of  ratios.  The  over-all  ratio  of  the  second  is  3.427 ; but  at- 
tempts to  express  the  proportions  of  details  by  means  of  subdivisions  of  this  rec- 
tangle have  been  unsuccessful.  If  a geometrical  scheme  was  used  by  the  potter, 
it  seems  likely  that  he  made  the  lekythos,  no.  175,  conform  to  it  exactly,  and 
changed  the  diameter  of  no.  176,  either  purposely  or  accidentally. 

The  ratios  of  no.  175  are  as  follows: 

Height 

Height  to  top  of  neck 

Height  to  bottom  of  neck 

Height  to  shoulder 

Height  of  lip,  neck  and  shoulder 

Height  of  lip 

Height  of  neck 

Height  of  shoulder 

Height  of  foot 

Diameter 

Largest  diameter  of  lip 

Smallest  diameter  of  lip 

Diameter  of  bottom  of  body . . . 

Diameter  of  foot 


3.309 

3.000 
2.500 
2.191 
1.118 

.309 

.500 

.309 

.1708 

1.000 
.5528 

.2764  = .5528  -f-  2 

.309 

.6584 


[ 220  ] 


LEKYTHOS 


[ 221  ] 


GEOMETRY  OF  GREEK  VASES 


177  Lekythos.  Inv.  95.49.  Beazley,  V.  A.,  p.  186.  Nurse  bringing  baby  to 
mother.  School  of  the  Meidias  Painter. 

Height,  0.1375  m.  Diameter,  0.082  m. 

The  containing  area  has  the  ratio,  1.691.  The  proportions  are  as  follows: 


Height 

1.691 

Height  of  band  below  figures,  bottom  .236 

Height  to  shoulder 

1.073 

Diameter 

. . 1.000 

Height  of  lip  and  neck 

.618 

Diameter  of  lip 

. . .4472 

Height  of  lip 

.309 

Diameter  of  shoulder 

. . .382 

Height  of  neck 

.309 

Diameter  of  bottom  of  body . . . 

. . .764 

Height  of  band  below  figures,  top 

.309 

Diameter  of  foot 

. . .7888 

[ 222  ] 


LEKYTHOS 


178  Lekythos.  Inv.  01.8119.  Two  youths  seated,  Eros  and  a woman  dancing. 
Free  style. 

Height,  0.132  m.  Diameter,  0.077  m. 

The  enclosing  area  has  the  ratio,  1.7071,  and  all  the  details  can  be  expressed  in 
terms  of  the  y/2  rectangle,  as  follows: 


Height 

Height  to  shoulder 

Height  of  lip 

Height  of  neck  and  shoulder . 

Diameter 

Diameter  of  lip  and  shoulder 
Diameter  of  bottom  of  body . 
Diameter  of  foot 


1.7071 

1.0606 

.2929 

.3535 

1.000 

.500- 

.7071 

.7677  = .3535  + .4142 


[ 223  ] 


GEOMETRY  OF  GREEK  VASES 


179  Lekythos.  Inv.  98.884.  Nike  cutting  off  the  forelock  of  a bull  in  the  pres- 
ence of  Athena  and  a youth.  Free  style. 

Height,  0.1515  m.  Diameter,  0.079  m. 

The  enclosing  rectangle  has  nearly  the  ratio,  1.8944.  The  height  to  the 
shoulder  is  1.118,  the  height  of  the  lip,  .382,  the  height  of  the  neck,  .3944.  The 
diameter  of  the  lip  is  .500,  that  of  the  shoulder  somewhat  less  than  .382,  that  of 
the  bottom  of  the  body,  .7236. 


LEKYTHOS 


180  Lekythos.  Inv.  95.1402.  Beazley,  V.  A.,  p.  186.  Acorn-shaped.  Eros 
and  woman.  Meidias  Painter. 

Height,  0.1585  m.  Diameter,  0.0595  m. 

The  containing  rectangle  has  the  ratio,  2f.  The  height  of  the  acorn-cup  is 
two-thirds  of  the  diameter. 


[ 225  ] 


GEOMETRY  OF  GREEK  VASES 


PYXIS 

181  Red-figured  Pyxis.  Inv.  93.108.  Hambidge,  p.  77.  (A)  Domestic  scene. 
(B)  Nike  crowning  a youth.  Free  style. 

Height  to  top  of  knob,  0.178  m.  Height  to  top  of  lid,  0.1365  m.  Height  to 
junction  of  vase  and  lid,  0.1205  m.  Diameter  of  lid  and  of  pyxis,  0.15  m. 

The  over-all  rectangle  has  the  ratio,  1.191.  This  can  be  divided  in  various 
ways,  one  of  the  simplest  being  .809+. 382.  The  lid  exactly  fits  a .382  (2.618) 
rectangle,  and  the  visible  portion  of  the  pyxis  therefore  is  enclosed  in  the  familiar 
shape  .809  or  1.236,  i.  e.,  two  whirling  square  rectangles.  The  area  in  which  the 
base  of  the  pyxis  is  enclosed  is  seen  to  be  composed  of  two  -y/5  rectangles.  The 
other  details  are  obtained  in  an  equally  simple  way. 

The  ratios  are : 


Height  to  top  of  knob  on  lid 1.191 

Height  of  knob  with  its  neck 2764 

Height  to  top  of  lid 9146 

Height  of  lid 382 

Height  to  junction  of  pyxis  and  lid 809 

Height  of  base 2236 

Greatest  diameter  of  lid  and  pyxis 1 .000 

Diameter  of  knob 2764 

Smallest  diameter  of  pyxis 750 

Diameter  of  foot 8292  = .2764  X 3 


[ 226  ] 


PYXIS 


[ 227  ] 


GEOMETRY  OF  GREEK  VASES 


182  Polychrome  Pyxis.  Inv.  98.887.  A neatherd,  a cow,  and  six  Muses, 
painted  in  colors  on  a white  ground.  Beazley,  V.  A.,  p.  128.  Hambidge,  p.  49; 
photograph  opp.  p.  50. 

Height  to  top  of  knob,  0.1765  m.;  to  top  of  lid,  0.125  m.;  to  junction  of  pyxis 
with  lid,  0.107  m.  Greatest  diameter  of  pyxis  and  of  lid,  0.146  m. 

The  interesting  relation  which  this  pyxis  bears  to  a white  pyxis  in  the  Metro- 
politan Museum,  New  York,  has  been  brought  out  by  Mr.  Hambidge,  op.  cit. 
Both  are  enclosed  in  the  rectangle,  1.2071,  i.  e.,  .7071  + . 500,  or  a -\/2  rectangle 
placed  horizontally  above  two  squares.  But  the  proportions  of  details  are  en- 
tirely different,  though  definable  in  both  cases  in  simple  areas  based  on  the  \/2 
rectangle.  In  the  present  example  the  lid  with  its  knob  is  enclosed  in  an  area 
made  up  of  three  \/2  rectangles  placed  vertically,  side  by  side  (cf.,  the  black- 
figured  kylix  by  Tleson,  no.  124).  The  visible  portion  of  the  pyxis  occupies  an 
area  composed  of  three  \/2  rectangles  placed  horizontally  above  two  squares. 
The  vases  with  the  lid,  but  omitting  the  knob,  is  placed  in  a rectangle  composed 
of  two  horizontal  v/2  rectangles  above  two  squares.  The  knob  with  its  neck  is 
placed  in  a rectangle  made  up  of  a v/2  rectangle  and  a square.  The  knob  alone 
is  contained  in  area  similar  to  the  over -all  rectangle  (1.2071),  as  is  shown  in 
the  drawing  by  the  diagonals  of  half  the  shape. 

It  is  noteworthy  also  that  the  top  of  the  lid  is  at  half  the  height  of  the  \/2  rec- 
tangle of  the  containing  shape,  and  that  a square  inscribed  in  the  centre  of  this  \/2 
rectangle  defines  the  smallest  diameter  of  the  pyxis.  If  a square  be  applied  at  the 
top  of  the  over-all  shape,  its  base  coincides  with  the  bottom  of  the  body  of  the 
pyxis;  and  the  indented  line  near  the  top  of  the  vase  is  at  half  the  height  of  this 
square.  The  intersections  of  the  diagonals  of  half  the  over-all  rectangle  with  the 
base  of  this  square  determine  the  diameter  of  the  base  of  the  pyxis. 

The  ratios  are : 


Height  to  top  of  knob 

Height  to  top  of  lid 

Height  to  junction  of  pyxis  and  lid 

Height  of  lid  with  knob 

Height  of  knob  with  its  neck 

Height  of  knob  without  its  neck . . . 

Height  of  base 

Height,  omitting  base 


1.2071  = 
.8535  = 
.7357  = 
.4714  = 
.3535  = 
.250  = 


.2071  = 


V2  + 1 
2 

+2  + 2 

4 

x/2  + 3 
6 

+2 

3 

V2 

4 

1 

4 

V2  - 1 
2 


1.000  = 1 


[ 228  ] 


PYXIS 


182 


Greatest  diameter  of  pyxis  and  of  lid 1.000  = 1 

Diameter  of  knob 2071  = 

2 

/Q 

Smallest  diameter  of  pyxis 7071  = — - 

Diameter  of  base 8284  = 2y/2  — 2 


[ 229  ] 


GEOMETRY  OF  GREEK  VASES 


PERFUME  VASE 

183  Perfume  Vase.  Inv.  97.367.  Covered  with  black  varnish.  A broad  band 
of  decoration  in  the  black-figured  technique  round  the  mouth;  a tongue  pattern 
round  the  lid. 

Height,  with  cover,  0.173  m.;  without  cover,  0.1285  m.  Diameter,  0.2395  m. 
Diameter  of  foot,  0.1475  m. 

The  containing  area  is  a 1.382  rectangle.  This  divides  into  two  .691  rec- 
tangles, each  made  up  of  a square  plus  a s/h  rectangle.  The  vase  with  the  cover, 
but  without  the  foot,  is  contained  in  the  two  squares;  the  foot  is  contained  in  the 
two  \/5  rectangles. 

Without  the  cover  the  vase  is  enclosed  in  a 1.854  rectangle,  i.  e.,  three  whirling 
square  rectangles  placed  side  by  side.  The  area  up  to  the  ridge  of  the  bowl  is 
composed  of  four  vertical  whirling  square  rectangles  placed  side  by  side. 

The  ratios  are  given  in  two  forms,  A,  with  the  height  taken  as  unity,  B,  with 


the  diameter  taken  as  unity. 

A B 

Height  with  lid 1.000  .7236 

Height  without  lid 7453  .5394 

Height  to  ridge  of  bowl 559  .4045 

Height  of  foot 309  .2236 

Diameter 1.382  1.000 

Diameter  of  ring  at  bottom  of  knob 1459  .0927 

Diameter  of  lid 618—  .4472 

Diameter  of  bottom  of  bowl 472  .3416 

Diameter  of  foot 854  .618 


[ 230  ] 


PERFUME  VASE 


183  A 


[ 231  ] 


GEOMETRY  OF  GREEK  VASES 


184  A 


184  Perfume  Vase.  Inv.  99.530.  Described  in  Ann.  Rep.,  1899,  p.  73,  no.  29. 
Hambidge,  p.  85.  No  figure  decoration.  The  surface  is  left  in  the  color  of  the 
clay,  or  painted  black,  as  shown  in  the  drawing,  in  which  added  red  color  is  repre- 
sented by  hatching. 

Height,  with  cover,  0.2455  m. ; without  cover,  0.1825  m.  Diameter,  0.2685  m. 

The  bowl  of  this  vase  resembles  an  archaic  Doric  capital  while  the  foot  recalls 
Ionic  column  bases.  The  proportions  are  heavy,  but  from  the  point  of  view  of 
execution  the  vase  is  a masterpiece  of  the  potter’s  art,  to  be  classed  with  the 
amphorae  painted  by  Amasis  (nos.  4 and  7)  and  the  kylix  made  by  Tleson  (no. 
124).  It  is  probably  to  be  dated  in  the  same  period.  Study  of  the  proportions 
shows  clearly  that  they  are  derived  from  the  whirling  square  rectangle,  but  many 
of  them  cannot  be  expressed  in  a simple  geometric  construction  or  in  familiar 
ratios.  The  vase  without  its  cover  is  contained  in  the  rectangle  1.472  (.6793). 
The  diameter  of  the  moulding  at  the  top  of  the  stem  is  .472;  the  height  of  the 
bowl  including  this  same  moulding  is  also  .472;  the  height  of  the  stem  is  .528. 
These  proportions  are  shown  in  the  second  drawing.  The  centre  of  a square  ap- 


[ 232  ] 


PERFUME  VASE 


plied  at  the  right  end  of  the  1.472  rectangle  fixes  the  diameter  of  the  moulding; 
and  a square  applied  to  the  top  of  the  .472  shape  at  the  left  fixes  the  height  of  the 
bowl.  Whirling  square  rectangles  applied  at  the  top  and  bottom  of  the  .472 
square  determine  the  height  of  the  raised  fillet  on  the  exterior  of  the  bowl  and  the 
level  of  the  bottom  of  the  incurved  rim;  the  intersection  of  these  two  horizontal 
lines  with  diagonals  of  the  square  determine  (with  a slight  inaccuracy)  the  diam- 
eter of  the  foot.  The  area  of  which  the  diameter  of  the  foot  is  the  long  side  and 
the  height  of  the  stem,  including  the  moulding  at  the  top,  is  the  short  side  is  a 
whirling  square  rectangle. 

If  a whirling  square  rectangle  is  applied  to  the  top  of  the  whole  containing 
area  including  the  cover,  its  base  coincides  with  the  uppermost  of  the  three  red 
rings  on  the  stem.  Its  diagonals  cut  the  line  of  the  top  of  the  vase  so  as  to  divide 
the  area  above  this  line  into  a square  and  two  whirling  square  rectangles  (4.236  = 
.236).  The  proportions  of  the  details  of  the  knob  can  be  simply  expressed  in  terms 
of  this  .236  square. 


[ 233  ] 


GEOMETRY  OF  GREEK  VASES 


184  A 


184  Perfume  Vase.  Inv.  99.530.  Described  in  Ann.  Rep.,  1899,  p.  73,  no.  29. 
Hambidge,  p.  85.  No  figure  decoration.  The  surface  is  left  in  the  color  of  the 
clay,  or  painted  black,  as  shown  in  the  drawing,  in  which  added  red  color  is  repre- 
sented by  hatching. 

Height,  with  cover,  0.2455  m. ; without  cover,  0.1825  m.  Diameter,  0.2685  m. 

The  bowl  of  this  vase  resembles  an  archaic  Doric  capital  while  the  foot  recalls 
Ionic  column  bases.  The  proportions  are  heavy,  but  from  the  point  of  view  of 
execution  the  vase  is  a masterpiece  of  the  potter’s  art,  to  be  classed  with  the 
amphorae  painted  by  Amasis  (nos.  4 and  7)  and  the  kylix  made  by  Tleson  (no. 
124).  It  is  probably  to  be  dated  in  the  same  period.  Study  of  the  proportions 
shows  clearly  that  they  are  derived  from  the  whirling  square  rectangle,  but  many 
of  them  cannot  be  expressed  in  a simple  geometric  construction  or  in  familiar 
ratios.  The  vase  without  its  cover  is  contained  in  the  rectangle  1.472  (.6793). 
The  diameter  of  the  moulding  at  the  top  of  the  stem  is  .472;  the  height  of  the 
bowl  including  this  same  moulding  is  also  .472;  the  height  of  the  stem  is  .528. 
These  proportions  are  shown  in  the  second  drawing.  The  centre  of  a square  ap- 


[ 232  ] 


PERFUME  VASE 


plied  at  the  right  end  of  the  1.472  rectangle  fixes  the  diameter  of  the  moulding; 
and  a square  applied  to  the  top  of  the  .472  shape  at  the  left  fixes  the  height  of  the 
bowl.  Whirling  square  rectangles  applied  at  the  top  and  bottom  of  the  .472 
square  determine  the  height  of  the  raised  fillet  on  the  exterior  of  the  bowl  and  the 
level  of  the  bottom  of  the  incurved  rim ; the  intersection  of  these  two  horizontal 
lines  with  diagonals  of  the  square  determine  (with  a slight  inaccuracy)  the  diam- 
eter of  the  foot.  The  area  of  which  the  diameter  of  the  foot  is  the  long  side  and 
the  height  of  the  stem,  including  the  moulding  at  the  top,  is  the  short  side  is  a 
whirling  square  rectangle. 

If  a whirling  square  rectangle  is  applied  to  the  top  of  the  whole  containing 
area  including  the  cover,  its  base  coincides  with  the  uppermost  of  the  three  red 
rings  on  the  stem.  Its  diagonals  cut  the  line  of  the  top  of  the  vase  so  as  to  divide 
the  area  above  this  line  into  a square  and  two  whirling  square  rectangles  (4.236  = 
.236).  The  proportions  of  the  details  of  the  knob  can  be  simply  expressed  in  terms 
of  this  .236  square. 


[ 233  ] 


GEOMETRY  OF  GREEK  VASES 


185  A 


185  Alabaster  Perfume  Vase.  Inv.  81.355.  An  inaccurate  drawing  is  pub- 
lished in  Journal  of  Hellenic  Studies  XXXI,  1911,  p.  87,  Fig.  15.  Provenance 
unknown. 

Height,  with  cover,  0.256  m.;  without  cover,  .1955  m.  Diameter,  0.1585  m. 
The  stem  and  foot  are  in  one  piece.  The  bowl  is  in  two  pieces,  the  upper  part, 
with  the  incurved  rim,  resting  in  a rebate  cut  at  the  inner  edge  of  the  lower  part. 
These  two  pieces  show  no  signs  of  having  been  cemented  together.  The  lid  and  its 
knob  are  in  two  pieces. 

The  vase,  with  its  cover,  is  contained  in  a whirling  square  rectangle.  Without 
the  cover  it  fits  the  rectangle,  1.236,  or  two  whirling  square  rectangles  placed 
horizontally.  It  is  noteworthy  that  the  most  important  horizontal  line  in  the 
composition  — the  ridge  of  the  bowl  — is  at  half  the  height  of  the  applied  square. 
The  drawing  of  the  vase  in  section  shows  that  the  junction  of  the  stem  and  bowl  is 


C 234  ] 


PERFUME  VASE 


at  half  the  height  of  the  whole  vase  with  the  foot  omitted.  This  drawing  also 
shows  that  the  lid  and  knob,  considered  separately,  are  enclosed  in  a square,  and 
that  the  joint  between  the  lid  and  the  knob  is  at  the  level  .191  in  this  square.  The 
knob,  therefore,  is  contained  in  the  rectangle  .809  applied  to  the  small  square. 

The  ratios  are : 


Height  with  cover 1.618 

Height  without  cover 1.236 

Height  to  ridge  of  bowl 1 .118 

Height  to  top  of  stem 882 

Height  of  bowl  alone 354 

Height  of  foot 146 


Diameter 1.000 

Diameter  of  rim  of  bowl 5528 

Diameter  of  top  of  stem 264 

Diameter  of  foot 528 

Diameter  of  lid 4236 


PRINTED  AT 

THE  HARVARD  UNIVERSITY  PRESS 
CAMBRIDGE,  MASS.,  U.  S.  A. 


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